This is classic "Nature versus Nurture" debate, and if you study psychology you'll never hear enough of this. The ultimate answer is that it is not one way or another, but rather a dynamic mix of both.
Tabula rasa vs. innate knowledge
The problem is first and foremost of definitions: how do you define knowledge? Proponents of tabula rasa suggest that we are all born with a "blank slate", a mind empty of all knowledge. But does that mean it's completely empty? Given our understanding of the mind today, the notion of tabula rasa is really only true if you put the emphasis on slate rather than blank—"blank slate" thus referring not to an "empty" state of mind but rather to a default state of mind. For example, the reason we are really good at learning languages is because our brain is shaped in a such a way at birth that makes it really easy for us to pick up words. Some people would argue that this innate ("default") skill at learning languages counts as knowledge, and as such the notion of tabula rasa would be false. But if you accept that an innate ability acquire language, for example, doesn't count as innate knowledge (perhaps because it is not factual knowledge), then tabula rasa is true. It should be noted that in the brain, skill knowledge is stored in a different location than factual knowledge (skill knowledge falls under procedural memory [dorsolateral striatum/basal ganglia] whereas factual knowledge falls under semantic memory [medial temporal lobes/hippocampus]). The procedural memory system is actually a lot better in many regards than the semantic system, which is why it's a lot harder to forget how to ride a bicycle than it is to forget the quadratic formula. Tabula rasa thus might be defined to only refer to factual knowledge (semantic memory) rather than skill memory, and as such it would fit with the prevailing scientific theory of mind.
Is the stuff we're born with the same as stuff we learn?
On the same note as above, we're really good at learning the world around us when we're very little. Few people realize the extent of the things we learn as newborn infants and how amazingly fast we learn it, completely unaided by anyone. For example, we are able to discern a notion of "space" and "distance" (that we live in a 3D world and things can be nearer or further from us, not merely bigger or smaller), of object occlusion and solidity (that objects don't simply disappear when they leave sight, that two solid objects cannot occupy the same space at the same time). Very early on, infants also develop a rudimentary understanding of logic, they that other people have minds that are different from there own, and that others can believe things which are false. The point is, we're good at these things because we're born with innate faculties which make it much easier for us. (Apologies, I would cite more of this but my child psychology books are at my parent's house. All of this stuff is heavily researched though, so you can easily find it online).
Okay... so what?
My point here is that humans are born with innate abilities to learn certain things easier and perform certain tasks better. The special thing about genes is that—outside of twins—everyone's are different. It is in fact very true that some people are born with better faculty for certain tasks than others. For example, I don't think too many people would dispute that some peoples of the world tend to—on average—be born with better genetic disposition to become great long distance runners than others (say, Kenyans). Just like some families have a lot of big people, others have a lot of small people, these kind of genetic predispositions which exist in the physical body also exist in the physical brain. Research heavily concurs that intelligence, for example, has a strong genetic component, but other aspects of the human mind do as well such as personality characteristics and even tendencies for things like alcoholism or depression.
So are some people born with better innate ability to draw or play music than others? Absolutely, in the sense that some people's brain structures are better suited to acquire and perform certain skills than others.
But no one is born already knowing how to draw perfectly. There is much experience that is needed, and training which helps develop the skill. The difference between a person with a genetic predisposition for drawing and someone without is that someone with the disposition would learn faster than someone without, and at more advanced levels may reach a level of proficiency that others cannot.
Great training and opportunity can often bridge the gap between innate ability and learned ability (although not always). But do not be misled into thinking that anyone of great skill automatically has great genes suited for the task. It may very well be the case, for example, that Beethoven worked long hours and studied extra hard to compose the music he did; he could very well have had poor genetic predisposition but practiced extra hard to be as good as he was.