The general consensus is to replace the original paper dielectric capacitors with the more modern polypropylene type. I went with polypropylene, but I've included both types in the parts listing in case you want to go with paper.
Note: If you are going to have the power supply open anyway, and you have an ESR meter handy, it is also recommended to check the ESR on all the capacitors.
250 VAC 0.01 uF PME 271 Y
17.9 mm x 10.46 mm x 5.06 mm, lead spacing: 15 mm
PME271Y510MR30 - Kemet Safety Film Capacitor - Paper - 0.01 uF - 250 VAC - Class Y2F17103101000 - Vishay Safety Film Capacitor - Polypropylene - 0.01 uF - 250 VAC - Class Y2
250 VAC 0.1 uF PME 271 M 610
24 mm x 13.8 mm x 7.2 mm, lead spacing: 20 mm
PME271M610MR30 - Kemet Safety Film Capacitor - Paper - 0.1 uF - 250VAC - Class X2
ECQ-U2A104KL - Panasonic Safety Film Capacitor - Polyester - 0.1 uF - 250VAC - Class X2
3SB 1-R - Time Delay / Slow Blow 1A Glass Fuse
3516P - Fuse Clips
Safety warning and disclaimer!!! Disconnect the power supply from the mains and leave it alone for 30 minutes before doing any work. Regardless, proceed at your own risk!
Disconnect the power cable from the motherboard by squeezing the left and right sides of the power plug to release. Remove the 4 screws securing the power supply, from the bottom of the Apple //e case.
Remove the 8 screws on the sides of the power supply (4 on each side) securing the bottom cover to the power supply.
From the top view the capacitors look good. No bulging. No tell-tale signs of rupture or leakage. We need to take the PCB out of the enclosure for a closer look.
A quick test with the multimeter indicates that the fuse is blown. That should be easy enough to replace.
Remove the 4 screws holding the power supply board to the power supply enclosure, one screw holding the ground wire to the chassis, and the two screws securing the power plug housing.
With the power switch still attached to the enclosure, gently slide out the PCB and lift it up and out of the enclosure. You should now be able to have a good look at the filter capacitors. These are not in great shape. One is slightly bulging already.
Use your trusty desoldering tool to remove the filter caps and the fuse.
Install the replacement parts.
I took the opportunity to install fuse clips. This fuse clip has two tabs. The outer tabs line up with the existing holes on the PCB; however they are larger than the existing holes (1.5 mm). I had to drill bigger holes (3 mm). I did not drill holes for the inner tabs - one of the tabs would need a hole that cuts one of the traces, and the other tab comes close to nicking another trace. Instead I just bent them flat and glued them on to the PCB with epoxy.
Keep the power supply disconnected from the motherboard and power up. A tick-tick-tick sound indicates a no-load state and is a promising indication of a working power supply.
Switch the power off, connect the power supply to the Apple //e motherboard, and power up again. You should be good to go.
tl;dr With a few cents and a little care, you can have your power supply up and running again!