A typical Presidential inauguration ceremony looks somewhat like this:
Obviously having tens of thousands of people in close quarters during a pandemic is not the best idea, but who will get to call the shots on how the ceremony is organized? Could Biden legally veto having a large gathering if he prefers having a small private ceremony instead?
There are two main committees responsible for planning the Inauguration and the surrounding events. The Joint Congressional Committee on Inaugural Ceremonies is a bipartisan committee composed of members of congress (from both houses). It is responsible for organizing the swearing-in ceremonies. The Presidential Inaugural Committee is formed by the president elect and is responsible for organizing surrounding events (balls, parade, etc.).
The Presidential Inaugural Committee is chosen by the president-elect, even though he doesn't play an active role in the planning. If Biden wants them not to schedule large balls or parades, he will pick people willing to do that.
The JCCIC doesn't work for the president-elect, but in principle the committee is a group of reasonable people working towards a common purpose. If Biden makes reasonable requests, in the name of public safety, about how the ceremony is conducted, the committee is likely to accommodate them.
Neither of these measures would directly prevent visitors from traveling to Washington on the day of the inauguration, but if there's not much to do, fewer people are likely to come.
