How safe is the Beechcraft 1900?
I would say that it is a reasonably safe aircraft, but, like most aircraft of that class, often operated in unsafe conditions.
The Wikipedia page does not have description for many of the listed accidents, but you can turn to the Aviation Safety Network which does.
There are 41 total hull losses listed there. From them:

Most common type of accident is controlled flight into terrain. That has nothing to do with the type and everything with operating to/from remote airports in mountainous terrain with limited choice of navigational aids around.
Then there are quite a few collisions, both midair and on the ground. That also has nothing to do with the type and much with operating at smaller fields often under VFR and without any radar.
Then there were several accidents where pilots mishandled engine failure. The aircraft is capable of flying on one engine and it had to be demonstrated during certification, but the pilots have to follow the correct procedure and may easily lose control if they don't. This kind of accident is more common in turboprops than in jets, but it does not seem to be more common in B1900 compared to other turboprops.
Only the three accidents that were contributed by improper installation of elevator or elevator trim control cables really have something to do with the specific type. They still have a lot to do with operating by smaller airlines and maintaining in smaller facilities with less experienced staff.

So no, the relatively long list of accident does not suggest any serious problem with the type.
Also, the accident rate and types are similar to the other types used in similar operating conditions like DHC-6, EMB-120, L410 or Do-228.