How to choose the impedance of a BNC adapter?
Most normal BNC terminated lab equipment uses 50 Ohm.  75 Ohm is mainly used for cable tv and related equipment.

Can you read the number printed on the side of the cables? For example RG58is 50 Ohm but RG59 is 75 ohm.

At low frequencies it does not matter although the amplitude may indicate incorrectly if you terminate in a different impedance from which it was designed.  For example if you use a function generator that was designed for 50 ohm and terminate it in 75 ohm the amplitude will 20% higher than intended (e.g. if you set it to 1V p-p output you will actually get 1.2V).

What constitutes low frequency depends upon the length of the cable - if the time delay in the cable is more than say 10% of a cycle of the highest frequency component then you may see ringing or other artifacts.  The propagation speed in a cable will be about 65% of the speed of light or about 1.5ns per foot.  For your 300MHz which has a wavelength of ~2 feet any impedance mismatch of more than a couple of inches could cause problems.