I lived in Aberystwyth for 7 years and know all the places you're talking about very well.
New Quay is nice, but will be full of tourists in the summer. I don't know what sort of cottage rentals you can get there, but there is a huge Haven park on the outskirts of New Quay. There is beach access from there, but it's down a steep trail with a very uneven staircase, so if you have the little one in a pushchair it will be a pain. Beach access from the centre of New Quay is easy enough, but it will be crowded. The beach is nice enough once you're on it though
Aberaeron is right on the coast, but again will be crowded in the summer. I've never been to the beach there os can't tell you wnat it's like. There's probably not enough to keep the kids entertained for a week though.
Aberystwyth has a lot to do for the kids and is obviously right on the coast. The beach is good and the Pier has an arcade in it with some kiddy style amusements and a great ice cream shop - I should know I used to make the ice cream for it. Again, I'm not up to speed on cottage rentals, but if you don't mind being a few miles from the coast, then Cwmrheidol is lovely. I'm pretty sure there was some holiday lets down that way - it's about 3 or 4 miles from Aberystwyth inland, down a lovely valley with the Rheidol river.
Borth is a dump, it's the drugs capital of West Wales and has absolutely nothing to do bar sit on the beach which is stony and pretty nasty. Actually, it also has the Animalarium which is a collection of animals that other zoos didn't want. It's expensive for what it is though, but entertaining enough. Just down the coast from Borth is Ynyslas which has the best beach in the area. There's no caravan park though, and again I don't know about the cottage market.
Hope that helps a bit.