The fact that Brownian motion is determined by such small particles and their interactions with each other, I would imagine that there is a probabilistic element to it, but for the most part, yes, it's deterministic. It's usually referred to as "random", but really it's just so complicated that we can't predict it. For example, no computer ever built by mankind has generated a random number. Instead, they generate pseudo-random numbers according to incredibly complicated equations. The effect is a set of numbers with no apparent relationship, but it's still deterministic.
So overall, other than quantum randomness, I'd say Brownian motion is deterministic, albeit super-complicated.