What background noise sound level should I have in my studio?

The lower the better!
It should be even lower in the control room, where we must be able to monitor any small imperfection in the audio tracks. Another point is that nowadays most audio equipment has large dynamic ranges, up to 100 dB. This means they can now work at their best capacities with a large sound level excursion. Remembering that we can listen to a sound pressure level of 90 dB only for a few hours (check the last number of Install pro to read more about it), we can get the importance to have the lowest background noise level: to be able to use the equipment at its best without damaging our ears, we must lower the lower limit to avoid raising the higher one.
Hence it is fundamental to silence at best hard disks, computer fans, audio equipment, and air conditioning system and maximize sound insulation with close-by rooms.
Many reference curves were studied to classify background noise: NC (noise curves) and PNC (preferred noise curves) are used in room interiors and we’ll show the more modern second type ones, just the ones that are related to studios.
Most of professional studios ask that background noise sound level never exceed the 15 or the 20 dB curve (i.e. to pass the THX certification one of the specifications a studio must have are to respect the NC25 curve)

Grafico NRC

In any project or home studio the background noise should never exceed 35 dB, but a roof of 30 dB should be kept as a goal. This is useful for example when buying any air conditioning machine, another advice is to think about a liquid cooling system for the computer and to make sound insulated boxes or boots for the noisier electronic or electric equipment (being careful not to over-heat them!).
The last limit should be used also in any recording room; the sensitivity of modern microphones is so good that their use would be uselessly limited otherwise.
We already said how all of these questions are as important in broadcast studios where background noise should not exceed 20 dB.