If a civilian is driving a car towards a person who is a police officer, that is historically sufficient justification for the officer to apply the instant death penalty against the driver. Even if a driver is trying to drive around an officer to get away, the officer's perception (or even a claim to have that perception) that the driver tried to hit them with a vehicle is enough to justify an instant death penalty against the driver. Based on these cases (there are plenty of them!), it would seem like it's never legal or OK to drive at another person, or in a direction that could be interpreted as attempting to hit another person. Given the severe & instant punishment, it seems like this is not even close to the boundary (hence the bold "never").
When a civilian intentionally drives into a crowd (e.g. in Las Vegas, or Isla Vista, or SXSW) and people get hurt (even if nobody's killed), that's considered a severe violation of the law and serious charges follow. Even driving towards a crowd, before reaching it, can be enough to warrant fatally shooting a civilian driver. This generally supports the conclusion above.
However, when it's the police doing it, it's fine for them to intentionally drive straight into/through a crowd that has lawfully gathered in a city's Market Square to watch Independence Day municipal fireworks (while it's not OK for a civilian to drive into a crowd gathered to watch municipal fireworks elsewhere). Driving directly into a crowd with loud sirens and lights is apparently a good crowd dispersion technique. The press doesn't even cover it except as a closing footnote on another story, to emphasize how hard the police are working to keep citizens safe.
When the crowd is not just assembled but also protesting, it's apparently OK for even non-police drivers to drive directly into a crowd. Here's one example from Ferguson, where the driver cleared it with police and even got charges brought against the protesters for damaging her car. Here's another example from Ferguson. Where it's someone driving into a crowd of anti-Trump protesters, police just shrug and help out the driver, before arresting protesters for being in the way. A driver in that situation might apologize, but doesn't face charges.
The clearest rules I can figure out from this is that it's legal to drive at people when it's the police doing it or when the crowd is assembled for something that might fall under a First Amendment category as opposed to people just being in a busy place. However, I don't especially like that answer. In trying to find a stronger one, my own personal sense of ethics clearly has no bearing or relevance to this question. If it did, all of the latter examples (even the police car) would be in the "illegal" category along with the first ones. So, what does the law say? What's the legal rule? Where does that rule come from (i.e. law citations)?