VARIAC: Procedure??? 2 old 30 wpc tube amps, unused for 4 years

4 years isn't all that long ago but along with a Variac you
may want to look into adding a dim bulb tester.

 http://www.geek-tips.com/2015/11/22/dim-bulb-tester/

moisture is not an issue, they are in climate controlled basement, thus no humidity, never too hot or too cold.

no diodes in the parts list.

It has one 5V4G Rectifier Tube.

so, others out, 5V4G in, do the variac thing,

then other tubes in (one 12AU7; one 12AT7; Two EL34 (originals were EL-37) and repeat the variac thing.

I'll go test the tubes now.

I agree with jea48 and totem395. Four years isn't very long really for new caps, I doubt you will have any problems. I have a variac and use it often on old equipment. For me, I wouldn't even use it for these Fishers if what you're saying is accurate. A slow voltage increase could create problems.

Never used one. Use a dim bulb tester on all new / old purchases.

So far, so good. 
Easy to build, easy to use.

This is the procedure for an H-K Citation 2,
a 60 W/channel power amplifier.You can adapt the steps for your own amp:
Powering up a long unused Citation II

1. Check continuity of all the grounds. Check there
are no shorts to ground from the B+, bias, or heater
supplies. Turn the bias pots fully clockwise. Center
the AC balance pots. Put a load resistor on each
channel's output (8 ohms or 16 ohms is fine, just
connect it to the corresponding terminal.).

2. Install the 12BY7A's - (line level tubes) DO NOT install
the power tubes yet. Bring the amp up on the variac 30
volts at a time. Be sure the bias voltage at pin 5 increases
along with the line voltage from the variac. Run the variac
up to 110 volts or so and you'll see 60+ volts at each pin 5.
Be sure all the tube heaters are lit once you get up to 90
volts or so.

3. Shut the amp down, let the charge bleed off the caps
(you may have to wait a while or carefully use a bleeder
to speed it up). Don't get bit by those caps - they'll knock
you on your ass!!

4. Install the remaining tubes (the power tubes). Bring
the variac up 30 volts at a time, watch for the bias voltage
to climb as it should and check the B+. Be sure all the tube
heaters are lit once you get up to 90 volts or so.

5. Once you get to normal line voltage on the variac,
then start biasing the power tubes. Start slow, set the bias
for each tube at the mid point of the meter (or about 50 ma).
Let it run there, watch for any problems, check voltages, etc.
Let it run 10 minutes, keep checking all the tubes. Then run it
to just below spec (a couple widths of the needle below
the mark on the meter). Do the same thing as before - give
it 10 minutes. Finally, run the bias to spec, and monitor/
readjust as needed for about 1/2 hour.

Once that's all done the AC balance can be set, and you're
ready to listen! But do check the balance often in the first
10 hours of use, then again around 25 and 100. After that it's
just once every 3-6 months.