At least in Northern Britain, the Romans did it by building a massive wall. Hadrian's Wall has a fortified gate every mile (modern archaeologists call them Milecastles), which makes the structure less useful for military defence but much more useful for controlling movement across the frontier. It's likely that there were soldiers stationed at these gates, who presumably checked what people were bringing into/out of the province and then charged the relevant duty. Whether they flat-out refused people entry, and what grounds they would have for doing so, has to be left to speculation unfortunately - we don't have enough evidence to know for sure. However, there is plenty of evidence for the movement of goods and money across the frontier, which would suggest relatively free passage at least for merchants and traders.

I'm on my phone at the moment, so can't pull together a list of suggested reading just now. I'll try to get something here later today or tomorrow!

Edit: in the meantime you might find something interesting in this other post by me: https://www.reddit.com/r/AskHistorians/comments/6ie7gi/how_effective_was_hadrians_wall/