As you might imagine, this question comes up quite a lot on these boards. Do a title search with the words "which resort" and you'll see what I mean.
What it usually comes down to is each person saying "their" resort is the place to go. That's why most of us chose them as home resorts. Each has it's own flavor and positive as well as negative sides.
Since your OP asked about Saratoga vs Boardwalk, I'll stick to those two. You can see from my signature that I'm a BWV owner, and that's the one I'm recommending, but not because I own there (honest!). I think it's better for families with kids the age yours are.
I think the fact that you can walk to both Epcot and MGM is such a convenience when you have little ones. So easy to take those much needed afternoon breaks. While you can take the boat if you want, you don't have to wait to go to or from either park if you don't mind the walk, and have strollers for the young ones. (5 minutes to Epcot, just under 15 to MGM at a moderate pace).
You'll need a bus to get to MK or AK, but that holds true for Saratoga as well.
There are numerous restaurants on the boardwalk, as well as a sweet shop and bakery. There are souvenir shops as well as a shop that sells food and other supplies for the villas. They have midway games, and there are special performers on the boardwalk at night. (ie a guy on a piano bicycle) There's a large creative kiddie area by the pool.
Saratoga is brand spanking new, and there's certainly something to be said for that. (But don't worry, BWV is in no way old looking) And if you prefer being near Downtown Disney for shopping or clubbing, Saratoga is a good choice. If your kids were older and interested in Disney Quest or Pleasure Island Saratoga might be better. But for toddlers I think BWV is the way to go.
I'm sure you'll have a great time whichever you pick.
