Hot Tub/Spa 70A Subpanel/Electrical Disconnect Box Wiring
You'll need conduit for the whole run, but that means you can use 6AWG wires

You will need to run conduit for this entire run; loose wires by themselves just won't do as a wiring method (see NEC 300.3(A) for the reference on this).  However, this means you can use 6AWG THHNs (/THWN, since the stuff you get these days is all dual-rated anyway) for the hots and neutral as wires in conduit can use 75&deg;C ampacities when terminating onto suitably rated lugs (as present on all modern distribution equipment), 6AWG copper wire can handle 65A when terminated on 75&deg;C lugs, and NEC 240.4(B) permits you to "round up" to the next higher breaker size under most circumstances we encounter here.

As to what size conduit to run? I would use 3/4" ENT through the basement run through bored holes in the joists, and then use a LB and a short length  (&lt;24") of 3/4" Schedule 80 PVC at the outdoor run to make the turn into the disconnect box.  The wires exiting the disconnect, then, are run in 3/4" LFMC, using appropriate fittings at each end.