One of the primary appeals of pure bred dogs (at least for me) is that literally "breed true." Sure -- there's variation -- but for the most part, the size, weight, markings, and personality within a particular breed are fairly predictable. Not so with mixes. You don't know what you're going to get with a mix. You might end up with a pup that has inherited the most negative aspect of each breed.
In order for someone here to say whether a breed (or mix of breeds) would work for your family, you need to describe your family and living situation. Active/couch potato? Young/old? Apartment/acreage? Working hours? Every one is different. That's part of the reason there are so many breeds.
As for your mix -- Pits are great for some people, but you need to watch out for breed restrictions in some cities. And for me, I don't think a border collie would work well. I don't want a dog that might be smarter than one of my kids!