Small glitch I found:
While I was in the game, I was walking and then another window on my computer jumped in front of my game. My character was still walking, making the walking sound, yet when I came back into the game he was just standing there, walking in place, until I moved him, then he stopped walking when I stopped moving him.
Not exactly a game-breaking glitch, but still. I would suggest that if the game is not an active window the game should automatically pause.
Also, pausing does not pause the music. That can get really annoying.

Content adjustments I suggest:
1) Add joypad support. Seriously. It doesn't feel like an arcade when I use the keyboard.
2) Someone else said it, and I agree with it. Swap the jump and punch buttons. Perhaps the best approach would be to allow the player to customize controls. (Especially so if you enable joystick support, because I've got like 12 buttons on mine and sure don't want it set to just buttons one and two.)
*) If you don't see how you can allow such customization in keeping with a classic arcade feel, I would simply point out that actual arcades had a special button inside the cabinet that the owner could press to go into a special set-up mode where they can set things like how many credits the game needs, how many points a player gets for certain enemies, and etc. Most ROM emulators neglect this, but it does exist.
3) I'd like to be able to play as Alec OR Tadashi. I know this is getting a little outside the realm of how this era of arcade games were played, but seriously when I hit the 2P START button I should just be putting the second player (Tadashi) in, not putting in both Alec and Tadashi. Back in the day when the second player was just a recolor of the first player it was fine, but when your second player is really his own character, people often want to choose to play as him.
4) Seriously, what is the point of adding more quarters if I still get a game over screen when I'm out of lives? I don't get a chance to add more coins and continue?

And mentioned this briefly in the comment section, but I figured I could give more information here in case you just didn't know. This game can and should be available as an Xbox 360 Indie game. ANYONE can make and sell an Indie game on Xbox live; the game just has to run on XNA and there is a small fee to enroll in the “XNA creator's club” so that the game can be submitted. The Indie games are available on Xbox Live and can only sell for one, three, or five dollars.
The greater half of the Indie games on there are styled after classic games, so this game would fit right in. Since people might be playing it while unfamiliar with the comic, you might want to include the comic's URL in the game.

Okay I have also found a glitch where the music kept playing after I hit game over.

I was playing a two player game so I could play as Tadashi while Alec was unmanned. I was on the second stage (Alec didn't make it past the first) and I got shot by a bullet that I ducked a little too late. The background music is just still going…

I started another game and it went back to normal.

Marscaleb, thanks for all this great feedback. I was fixing bugs tonight for a new release (v1.1) and I've found your suggestions very helpful.

The first glitch:
This is a glitch that I probably won't be able to fix. It's lucky that it's not a very serious one. The Game Maker software that I'm using doesn't seem to have contingencies for when Windows swaps priority to something else.

Your various points:
1) I'm actually looking into this right now. However, one of the problems is that everybody has a different joypad, so it's highly likely that if I do add joypad support, I would have to assign action buttons to button1 and button2. But if you're really itching to try the game with a joypad, there are numerous programs available that will map your joypad buttons to keyboard keys in whatever configuration you like. I've played with this one, though I still prefer the keyboard. Nevertheless, I'm looking into joypad support if it's not a pain to implement.

2) I designed the game after many of the old arcade games of the early 80's, many of which tended to have the jump button first. In addition to this, the button configuration is so hardwired into the coding that it would take me all day to change…. However, I'd consider it if I can find a way to remap buttons in a custom manner.

I love your idea of simulating the arcade dipswitches. If I figure out a way to accommodate button remapping, that sounds like a fantastic way to do it.

3) That's a perfectly reasonable request. It's a hard one to pull off considering the way the game is put together, but it's one I'll see if I can do for a future update.

4) As with #2, the game is designed to be like an early-80's arcade game, so it's not really designed to allow you to continue. Of course, this does make the game tough, but with a little perseverance, you'll be able to see all seven stages. I know it's a bit of a pain, but conquering that 80's-style difficulty is part of the nostalgia trip and fun. When you unlock bonus characters, it's all the more rewarding.

XNA: The problem with getting my game onto another platform (be it XNA, PSN, DSiware, iPod, etc.) is that I would need a way to convert it from Game Maker format. I don't know how to program… Well, I know Qbasic and some C++/Java, but I'm so far from being a game programmer that there's not much hope. If I find a reasonable way to convert the game to any other platform, I'll be sure to jump at the opportunity.

The second glitch:
That's an interesting one I didn't know about. I'll try to reenact the glitch myself and see if I can fix it. Thanks for pointing it out!

Well… I know it sounds extreme but there is the option of completely re-writing the game in XNA. It's not as hard as it sounds at first, trust me. Of course it still is time consuming, but theoretically it would be worth it when you can actually make a couple hundred dollars from selling your game.
If you really want to save time then you could asking your readers if there is a programmer who is willing to help.

Anyway, while I completely understand and in fact that you are deliberately trying to emulate the classic 80's arcade games, it still is not the 80's right now, and there is really no reason to bind yourself to a less-desirable form.
What I mean to say is this. Those who truly want to the classic arcade experience can have the classic arcade experience. Those who are not so interested in it shouldn't be bogged down and forced into it. There are the hard-core retro gamers, and then there are just people who like to play the game. There is no need to punish either party; it wouldn't be hard to let your game have both options.
Maybe you can have the default arcade version of the game, and you could have the “Home console port” that gives you continues. (Just off the top of my head.)

Oh and one more thing about joypad support…

When someone runs an arcade game on their computer (which is what ideally this game is emulating) the emulating software usually has options for what buttons on your computer emulate what buttons on the arcade. So setting up a joypad could easily be setting it up through a fake emulator option menu, and still be matching the theme and back-story you have made for this game.
Future updates and major changes could also be described as user-created hacks that the emulator runs.

Also I just found another glitch. It seems that accidentally pressing Alt will just f*** your game up. I did it while I was moving and Alec just got stuck moving into the edge of the screen. It also causes the game to stutter.

Marscaleb, thanks for sticking with me on this.

It's okay if you're not as into retro gaming as the game might expect you to be. I'm making some changes that should cater to those who want a slightly newer experience. I don't think I'll budge on the game's difficulty or allowing continues, but you should expect these in an approaching update:

I've figured out how to change the keyboard mapping. Because of this I've made it so that you can switch the Punch/Jump buttons from one easy setting. I've even included an option to switch P1 and P2 controls entirely. I'm aware of the button configuration options of programs like MAME, but they're harder to pull off than you might realize. I'm hoping these added options in v1.1 still have the desired effect.

Also, you can start as Tadashi in a one-player game now. It's kind of a hack-solution, since you're really starting a two-player game with Alec already dead, but it's a lot better than having to kill him off yourself just to play as Tadashi.

Last of all, I almost have joypad support figured out. The gameplay works almost perfectly, but I'm having some trouble with interpolating diagonals. I'd like to be able to make the buttons customizable, but if I can't figure that out, the controls will default to button 1 and button 2 of the controller. I can't plan for every USB controller out there.

As far as the ALT bug goes, I don't think that's a bug I can fix, since it's part of the way the program works. But I'll still look into it.

If you can't have the game dynamically check the button possibilities of what joypad is plugged into the system, then perhaps you should assume that the joypad can support at least up to four buttons. In all honesty, I do not believe that I have ever seen one that does less than that.
So, have the joypad button mapping work with buttons 1, 2, 3, and 4, and let the player assign the jump and punch to any of those four.

At any rate, please don't forget to make sure that you can use a gamepad AND the keyboard, in case folk with one gamepad want to play multiplayer.

Playing as Tadashi will just kill of Alec?
Won't that still lead to having the “who did better” screen at the end, which screen never shows Tadashi if you play only as Alec? I mean it's not like that's going to break the game or anything, but it would look a little unprofessional.
I can't exactly look through your code and tell you how it should be done, but in all honesty the 1p and 2p starts should just be a Alec Start and Tadashi Start, and if both are hit then both players are put in the game.
…come to think of it, I can't recall any arcade games this old having multiplayer, but the oldest ones I can think of worked that way. Of course, they also gave you a few seconds to add more quarters for more lives. I guess your retro appeal is aiming for something just a few years too old to appease me. :)
(This game *did* come out the same year I was born.)

Anyway, I do actually love your game. Have I ever mentioned that?