While each senator is probably fairly well known in their own state, to achieve national recognition they'll need to do something really big, really great, or really stupid—and sometimes you don't have any control over the last one (you know who you are...Todd).
Fame is short-lived, and it can be a fairly dubious goal for a senator to aim for. Some people, like comedian Al Franken and astronaut John Glenn, were already famous well before they hit Washington.
If you want to hold public office because you want to be famous, you probably should move on to the next table at your career fair. The People want you to do this job for them, and if you're really just doing it for yourself, they won't be happy.
Senators don't need to be famous. Just ask former Senator Robert Byrd. Born in 1918, he ended up serving a whopping fifty-one years in the U.S. Senate, from 1959 until 2010. They called him an "institution within an institution." He has numerous buildings named after him around both Washington, D.C., and his home state of West Virginia.
And how many people outside of D.C. or West Virginia have heard of him? Well, now you have, so there's that.