It's both. Most successful breeders will breed horses with the traits they want to another horse with traits they want. In cross breeding you'll look to avoid stallions or mares that have undesirable traits even if they have some you want, because you'll always have a 50/50 chance of it being passed on.
Check out both the stallion and the mare and give the same weight to both.
That's why developing a "breed" takes so long. There's a fair ammount of culling that goes on in the beginning to remove horses that might have something unwanted out of the gene pool during the developement.
Of course the problem becomes when what the breeders (new breeds and especially older ones) want is so specific and for a money driven industry that they'll keep breeding for the desired traits and not worry about some others that would normally be less desirable (especially if one was at the expense of the other). TB are a good example of the latter. Todays TB are generally faster, but not as sturdy as the TB of 300 years ago.
Nothing is certain. My older mare looks a lot like her father, but has more of her mother's personality, height and lead mare attitude. My younger mare took absolutely nothing from her mother and with the exception of her star, strip and snip is virtually a carbon copy of her father in every way.
It's important to see what you want in both parents.