In the "non-fighters" category (since I dont get into those at all):
1) Bubble Bobble... hands down the best. It's cooperative only. You really can't hurt the other player
... also... Any BB clone like SnowBrothers or similar. 100 levels and even with 2 players and unlimited quarters on the REAL arcade board, it's very hard to get past level 75 or so, at least we never have.
2) Mario Brothers (not Super)... The best before bubble bobble came out. It's a great game that can be played either cooperatively (though it's easy to make mistakes and hurt the other player) or as a rather venimous against each other game..... hint... get rid of the POW before leaving the bottom row!
3) Joust (or Balloon Fight on PlayChoice-10...basically the same idea).
If you like fighters, then go with Simpson's. It can do up to 4 players, is basically identical to Double Dragon, but much more cartoony/a lil bit less "real violence" for the youngsters.