All of Europe has affordable places. You need to be more specific. Have you thought about driving and camping? All affordable then, but do you want beaches, activity holiday, culture and history? Hotter weather, cooler weather? City or rural? You can camp in all these sorts of places.
Or do you mean where can you get a luxury AI for less? Hard to tell.
From me - if you are prioritising easy to plan, easy to get to, not too hot, not too cold, lots of activities for kids both within the accommodation and out and about, you can’t beat camping in the Dordogne. There are lots of campsites with lakes for swimming, organised kids clubs, lots of other kids running around for yours to make friends with, and you can hire a tent or mobile home if you don’t have your own tent. Food is cheap both for self catering and eating out. Activities outside the campsite include everything from Go Ape style tree climbing routes, canoeing down the river, visiting caveman caves, medieval castles with catapult museums, gardens with mazes and via ferratas. It’s lovely.
If you want cheap AI and just cooking on a beach, Turkey is a great option. There is sea, sand, it’s cheap (although not as cheap as it was.) Quality is variable though, but your main specification was for affordability and you didn’t mention quality at all. We’ve stayed in mobile homes again here, cheap as chips and desert island beaches, but we were never looking for cheap hotel and kids club.
If you tell us what your idea of a great holiday is, you might get better suggestions.