Peasholme park is lovely.
The Sea Life Centre at Scarborough is good, obviously there is the beach, with all the tat shops on the front like you get in any seaside town.
Bridlington has a nice fair on the front - not too expensive.
Whitby itself has donkeys on the beach, and sometimes a bouncy castle, there is a bit of a fair area, but tbh it's not great for littlies.
Filey is nice if your kids are still in a pushchair - it's too hilly for just walking children but again, the beach is okay and there are some nice things to do on the front - crazy golf and stuff.
Out of the four, Scarborough has the best shopping centre.
Robin Hood's Bay is gorgeous and well worth a visit if you are in the area, but a day trip is probably all you need.
Also Cayton Bay is spectacular so you must go and have a look there.
Scarborough has a nice little sort of theme park across the road from Peasholme Park which I've never actually been to, but I believe it is good for the kids.
If you have a spare few minutes, it's worth ringing around some of the holiday parks and seeing if they allow you in to use their facilities if you aren't staying there - some of them have fantastic things for the kids to play on.