William H. and Ellis L. Baker began production of shotguns about 1878, Later, L.C. Smith bought out the original company. About 1888, the Hunter Brothers bought the name and made "New Baker" guns. If yours is marked Baker Bros., and has no L.C. Smith markings, it would appear to be from the first production.
The guns were good, solid guns, but not classics like the Parkers. Value depends primarily on condition and could go $1000 in top condition. Problem is that there is not a lot of collector interest in shotguns other than the classics.
You have been told correctly not to shoot it. It is best kept for sentimental reasons.
One minor "tribute" to the Baker was the number of cheap shotguns marked "BARKER", which was intended to confuse the buyer into thinking the gun was a Baker or a Parker.