It varies greatly depending on your field. If you study the US during the mid 19th century, you can focus on a lot of other things and only ever really have a cursory knowledge of art. Obviously, art can always provide information about a time and place, and major works of art ate important touchstones, but overall you can do a lot of your work from written records of events if you're doing something like antebellum politics or something. You should probably be aware of important and popular imagery, but you won't work with it regularly.

Switch to something more like my focus on Ancient Persia, and its very different. Really this is more true the further back in time you go too. In times that we have less written evidence for, historians need to interpret historical information based on physical remains, which includes artwork. The style of art, what it depicts, and how it was used can tell us a lot about where ancient cultures came from, how they lived, or what they valued. That's very useful in lieu of written explanations.

The one exception I'd make to the "art is more important as you go back" rule is the very recent past. If you study 20th century history, especially post-WWI and in western Europe and the US, then you're also studying the history of communications technology. The massive advancement from the telegraph in the late 19th century to the cell phone in the early 21st allowed for culture to spread and change quickly and things like art and music became more entwined with major events and cultural movements that a historian of the recent past might focus on, even if they don't focus on art.