Can the government stop funding to an artist or museum?
It depends. Generally, the government may not prohibit speech because it finds the speech to be offensive see Texas v. Johnson, 491 U.S. 397 (1989). Government may decide not to fund the arts at all, but if it does fund them (for instance, in the form of support to a museum), it cannot single out works to be banned or removed because they may be shocking or offensive.
The First Amendment protects also against viewpoint discrimination suppressing certain points of view in favor of others. However, if the government promotes a program and defines the program’s limit, it can fund speech that promotes its goals, even to the detriment of other goals, as seen in Rust v. Sullivan, 500 U.S. 173 (1991). In the case of NEA v. Finley, the Court did not see any harm in adding a decency standard to an already subjective process. Therefore, the NEA can take into account whether a piece of art offends the sensibilities of the public when deciding whether to fund the art.