If you are asking if the person has ever been to any museum, use "a." If you are asking about a specific museum that you've already mentioned, use "the."

You would say 'Have you ever been to a museum?' if you want to know if the person has ever been to ANY museum at all in their lives. As most people have been to at least a few museums in their lives, this is something that you might say to a child. Or if you meet someone who has always lived in a tiny village and never been to a city, then you might think that they have never been to a single museum in their life and you would say 'Have you ever been to a museum?'. You would say 'Have you been to the museum?' if you are talking about one specific museum which both the speaker and the listener know about. If there is one museum in your city, you might say this to a tourist to find out whether they have been to this particular museum or not. NB Please pay attention to capital letters and punctuation.

Rachel: "What are the plans for tomorrow?" Emily: "Would you like to go to A museum?" Rachel: "I took my grandson to THE Museum of Science recently, I don't want to do that again so soon." Emily: "I was thinking of THE Peabody-Essex Museum. I love that place, it's not A huge museum like THE Museum of Fine Arts, but it's a gem. Rachel: "Let's do it. What time do you think we should go to THE museum?" Emily: "THE museum opens at 10 a.m., so let's leave at about 9."