A President doesn't fund the government, the US Congress does. Failure to fund by Congress results in a shut down due to lack of funds. A President can veto an appropriation bill, and it goes back to Congress, who can fix the bill so the President will sign it, over-ride the veto with a 2/3 majority vote and fund the government, pass a continuing resolution to fund the government at existing funding levels until a fixed funding bill is ready that the President will sign, or do nothing and shut down the government. Under any scenario, the decision to fund or not to fund is made by Congress, not the President.

But I understand why reporters and people refer to the veto by a President as "shutting down the government", even though that's wrong, because the funding doesn't die there, it goes back to Congress for action.