Dont forget, you are listening to an old dirty analog audio source. this isn't the clean crisp audio you'd expect from a digital MP3 player or CD by any stretch of imagination. it's like the people who complain here about how blurry the image is on their CRT monitors when they've been staring at a 1080p HD video screen for the past 10 years. it's going to be muddy, and dirty looking.
that said,
the capacitors in the amp likely need to be changed. they help filter and eliminate noise in the power system that can manifest itself as humming through the speakers. At this point in time they are quite old (getting on 40 years) and really, quite honestly, only last a few years at peak performance. (capacitor ratings are measured in a few thousand HOURS of run time before going out of their rated specification) if the are still original from the factory, they a far past their prime. it can be an expensive proposition to replace them all, but in the long run, this is going to be your best bet...and in a home environment probably would last another 40 years.
in the mean time, before jumping into spending a bunch of cash... you can double check the power and grounding wires around the machine. look at your line cord...has the ground been busted off so someone can plug it into a 2 prong outlet? is the outlet you have it plugged into properly grounded? get one of those outlet testers with the 3 lights on it and see if the outlet is wired properly. a reversed outlet is not only dangerous, it causes a boatload of noise issues as often the lead designated a neutral is tied to ground. in which case, your live may be tied to ground.
once you know the power in is good, you can work inside. Lamps and everything. a loose, dirty, or corroded connection in the power system can introduce lots of noise into the power system and ultimately the audio section that the old capacitors cannot filter out. unplug everything and plug it back in. this will help get a good connection back. any screwed down grounds that look suspect, pull them off and clean underneath with some steel wool or a small wire brush. make sure to screw it back down nice and tight. you want it to bite into nice clean metal. even if there is only a couple small things you have found, a bunch of tiny things can add up to a big thing.
also check out the shielding on the cartridge wire. make sure it's not broke and is connected to ground good. check the player itself and make sure it's ground is good and tied to ground properly as well.
you may not eliminate noise completely. there will always be some hiss, or hum, or something. it's part of the way these systems work. but you can give yourself the best experience by making sure everything is as tight and clean as you can get it..