The ability to sing in tune is a major asset for an actor.
Even if you have no desire to perform in a full-blown musical, the necessity to sing crops up in the most unexpected places. Shakespeare is perhaps the most obvious example where an actor might be required to sing for his supper.
Would you really want to turn a part down simply because the prospect of singing in public leaves you in a cold sweat.
Or more likely you would take a part and suffer the most crushing anxiety perhaps to the determent to your role as a whole because of this one weak area.
Singing can be taught.
You may never turn yourself into a Pavarotti but you should be able to get to the stage where you can "put a song across" with confidence.
The value of a good singing teacher cannot be over stated.
Next on the list of importance is practise, followed by a good knowledge of the how's and why's of singing and the voice.

Tips and hints
1) Warm up your voice each time you sing.

2) Practice at least every other day. Like any muscle, the voice needs to be worked
to keep it in good order.

3) Drink plenty of water, before, during
and after singing.

4) Never force a note, you'll increase
your range by lots of gentle
practice.
You will damage your voice by
forcing.

5) Dairy products coat your voice and make
your cords tighter, avoid them
before singing.

6) Be confident, nerves need to be controlled, learn to relax.

7)Know your voice, your range and your limitations.
Choose your songs and keys accordingly.