Can I install a 60 amp breaker for a 30 amp disconnect with 20 amp fuses?
The breaker at the panel protects the conductors leaving the panel. The fuses in the disconnect protect whatever is downstream of the fuses. Those are two separate jobs, and a 60 ampere breaker does not become acceptable just because there are 20 ampere fuses somewhere down the line.
Section 240.4 requires conductors to be protected against overcurrent in accordance with their ampacities. If you run 10 AWG copper from a 60 ampere breaker to a 30 ampere fused disconnect, that 10 AWG is protected at 60 amperes, which violates 240.4(D)(7) limiting 10 AWG copper to 30 amperes. The fuses in the disconnect do nothing for the conductors upstream of them.
Two ways to make it legal
The first way is to size the conductors between the panel and the disconnect for the 60 ampere breaker. That means 6 AWG copper at 65 amperes in the 75 degree C column of Table 310.16, or 4 AWG aluminum. Then land those 6 AWG conductors on the line side of the disconnect, put 20 ampere fuses in it, and run the smaller conductor from the load side of the fuses to the equipment. The 20 ampere fuses now protect the small conductor legally.
The second way is to use the tap rules in 240.21. Section 240.21(B)(1) allows a tap up to 10 feet where the tap conductor ampacity is not less than the calculated load and not less than the rating of the device it supplies, the tap does not extend beyond the equipment it supplies, and the tap conductors are enclosed in a raceway if they leave the enclosure. Section 240.21(B)(2) allows a 25 foot tap where the tap conductor ampacity is at least one third the rating of the upstream overcurrent device. One third of 60 amperes is 20 amperes, so 12 AWG copper at 25 amperes clears that threshold, and the tap must terminate in a single device rated no more than the tap conductor ampacity.
Why the disconnect rating matters too
A 30 ampere fused disconnect has a 30 ampere switch rating and 30 ampere maximum fuse clips. Section 110.3(B) and 404.14 require the switch to be used within its listing. You may put 20 ampere fuses in a 30 ampere disconnect, and you may not put 60 ampere anything in it. Feeding a 30 ampere rated switch from a 60 ampere breaker also raises the available fault current question under 110.9 and 110.10, though most 30 ampere fused disconnects carry a 10 kA or 100 kA series rating that covers typical residential and light commercial values.
The simpler answer for most jobs
Put a 30 ampere breaker in the panel, run 10 AWG copper, land it on a 30 ampere fused disconnect, and size the fuses to the equipment nameplate. Everything is protected at every point and no tap calculation is needed. Use the 60 ampere breaker only if you genuinely have a 60 ampere load somewhere else on those conductors.
Field takeaway
The breaker protects the wire it feeds. A 60 ampere breaker requires 6 AWG copper to the disconnect unless you can document a 240.21(B) tap. Fuses downstream never fix an oversized upstream device.