Back in the day we either had infinite continues or a limited amount on the consoles. In arcades our continues were only limited to how many quaters we had. So if you were playing Ghosts & Goblins and you only had 14 quaters, you were screwed.
Depending on the game, I say it's really nice to have unlimited continues. With certain SHMUPs, I think it really helps to have unlimited contines. For a game like Zanac, where the levels are long and there are 12 levels total, those infinite continues will really come in handy.
And as I already mentioned with the Ghost & Goblins games, if you didn't have infinite contines, you may as well pack your crap and go home.