I would really want to know about prior visits, the number and ages of people in your party, dining plans, etc. before I'd answer. Also, if you're a balcony person or not.
I've stayed at YC and BC (and BCVs) with young kids. We have always gone during lower-occupancy times of year, so a less-crowded SAB has been a big draw for me and my kids. We liked having Hurricane Hanna's nearby for QS food (both by the pool and to take to our room), we used the QS place at BC a few times (YC has one now as well), ate at more casual restaurants and lounges at both places...and we only walked over to the BW area a few times over three trips for dinner. We are also big Epcot fans, so being near to the WS for dinner was great (though I realize BWI is close to that as well). We loved being able to walk to the lighthouse to catch the boat to the Studios. We loved the look of BC's latest renovation and felt the room was luxurious. I like the look of YC's renovation, but our stay there was in the old style rooms. During all stays in the YC/BC complex, we asked to be close to transportation and were very close to elevators to the lobby.
Regarding balconies, twice we had full ones, once a Juliet balcony. We could see Epcot fireworks from both full balconies (one a standard view, one a garden view. Ironically, our Juliet balcony in BC was a lake view and the most expensive...the view was lovely but we couldn't see fireworks). We were ok with the Juliet balcony but really prefer full ones.
However, now that my kids have experienced SAB, they're ok with staying at BWI and trying the clown/roller coaster slide. I've heard the pool at BWI is also a bit calmer than SAB overall (a big draw for me). We'd likely get a full balcony (I'm not sure if there are a few without). I really like the look of the rooms, though I've heard some of them can be particularly far away from the center of the resort. We would probably miss more QS options as well as indoor TS options at BWI compared to YC/BC. The restaurants in the general BW area aren't our favorites (or at Swolfin). So overall, we'd like to experience BWI b/c it's 'new' to us. But as you can, for a variety of reasons, it wasn't our first choice of Epcot area resorts.
I hope that helps
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Edited to add: you edited your post as I was typing! If your kids like pools (and if you're going when you'll have good pool weather), I'd probably chose either YC or BC given the ages of your kids. SAB is a treat for kids who like swimming...there's a great, shallow sandy area for young kids and the older ones will like the lazy river and slide. Kid's meals at HH come in buckets to play in the sand after. If you want a balcony, go with YC...if it's not so much of a concern, go with whichever decor rocks your boat the most.
One more note...the lobby at BC is busier and often louder than the lobby at YC due to the presence of the Cape May buffet. BC's lobby is great for people watching, YC's lobby is much more peaceful if you enjoy a quiet break. Some say YC is less 'kid friendly' because it's more sedate and formal, but my boys loved it and the nautical theme, fwiw.
