BLUEMOMENT

Got a bilge alarm? Just a loud bleeper wired across the automatic pump in it's basic form. If you have, you probably (hopefully) don't know whether it still works. Now is the time to test it as you empty the freshwater tank to avoid freezing or lighten the boat. Tip a few basins full into the bilge until it goes, or - if it doesn't, add the job to the list. I can assure you it's good to get early warning of anything going wrong in the nether regions and like any other safety device, you need to have confidence it will work when needed.
You will then have a little fresh water in the bilge to do the cleaning once ashore .

Or just put a wet cloth across the contacts?

Contacts - if you could reach them, possibly. However ours is a barometric switch* connected to a belljar sort of thing and needs a bit of PSI to operate. The up side is that all the electric stuff stays out of the water (unless something awful happens)
* The switch was originally the level sensor in a washing machine

My bilge is never really dry - should it be?

In a Rival 32, the bilge has a sump, which is actually a void in the encapsulated keel area. The lead keel occupies most of the hole, but there's a void about 250mm square, but with one side to a point facing aft, all inside the fibreglass encapsulation of the keel. Its quite deep, perhaps 750mm deep from the floorboard? The bilge pump hose 50mm dia doesn't ever pump out all the water, and there's usually about 50mm left in there.
But various things leak a little, and let a few bucketfulls in there over time. Mostly this is fresh water, rain into the cockpit locker tops, a few drip for this or that, and so it accumulates.
The windlass brings in a little.
Cleaning the log impellor (on the run, in the water) brings in sea water.
5-10 pumps on the pump handle every so often clears it out.

Should I expect better in a 40 year old boat?
Should I put in more effort to stop these various small topsides leaks?

No Geoff, (in my opinion...) The beauty of a nice deep bilge is that any ingress is 'managed' - ie. you know where it is, if not where it came from. The addition of an audible alarm helps keep quantity in control if it ever gets to that stage. My stuffing box is drip-cooled, so there is bound to be some water down there.
The downside is that you can't reach to get it properly dry ( I have a sponge on a broom-pole!) so there is always a little that the pump can't clear and some that runs back down the outlet pipe.
As long as it's not allowed to stagnate or collect fuel, it doesn't seem to affect the quality of life aboard.
I created a wee catchment area round the log impellor so that the mugful could be easily captured and mopped up. This is it before glassing and danbolining.

I tend to think that way too. As there's always a little water, and sometimes some to pump out, I lift that floorboard every day and take a look.
I have had a little diesel get in there (changing fuel filters, and once removing and then bleeding injectors - at sea), so on these occasions I've squirted detergent down the hole, tipped in a kettle of hot water and then pumped it out. Hardly a slick! perhaps a barely visible sheen astern!

My technique to clean the log impellor involves using the floorboard that covers it as a sheild (its in the heads, which is between main saloon and forepeak cabin) so i don't get too wet, have the blanking plug ready, pull the pin, pull the impellor, stem the fountain with the blanking plug, and set to with a toothbrush. then reverse procedure. On a good day perhaps a litre spills. On a fumblefingers day maybe 5 litres. It flows along the keel-line to the sump, slowly, but maybe that helps clean out the drain holes?

I've noticed that the application of a little grease to the blanking plug, then later to the log impellor unit, makes all this so much easier, eases those O rings into the hole. They say that lubricating your plug before putting it in the hole is always a good idea.!