I use both FB and twitter for work. As others have said you can keep the privacy settings tight and have a work email so everything is fenced as it were.
I also think as a musician you have a great opportunity to create a 'profile' that enhances your work. On twitter if you retweet or follow certain people, events, organisations you start to build a profile that represents your place within the community, your interests, your passions etc. I think it can be hugley positive. FB is a bit more work in my opinion, but good for events, to promote if you need to sell tickets etc or advertise teaching.
You can use social media software that posts automatically, so that you don't have to be present every day to keep things up to date, because social media only really works if you drip feed a steady input of content. So posting once a month isn't going to really benefit you - you will get 'lost' in the sea of 'stuff'