At bases only, supplies may be destroyed by air attack (unlike against units or stacks in non-base territory) or by surface fleet bombardment. When you see "airfield supply hits" or "port supply hits" you will see your supplies deteriorate substantially, sometimes alarmingly. In many cases you can ameliorate this in advance by dumping enormous amounts of supply at bases that you expect to be besieged.
So it is true that bombing the airfields do lower supply,
however it is not true that is caused by 'repairing the bases' but instead by the loss during the attacks.
That could have been an assumption error, the reason for supply loss.
Does bombing port or airfield hit more supplies then 'ground attack' method?
Bombing the AF MAY lower supply. Usually does, but may not depending on lots of factors.
Repairs to AF, ports, forts do not cost supply. Moving between levels in construction does, at increasing, non-linear rates.
To answer your question from above, when someone says they "bombed Saigon", and it will cost supply to repair, they're talking about bombing industry: HI, LI, oil, refineries, yards, engines, aircraft, armament, vehicles, manpower. Repairing damage to them does cost supply. 1000 per point repaired and 10,000 in the hex to begin at all.