Is there any way a portal frame with pin joints at the top of the columns can be stable? Are portal frames ever done with pin joints at the bases? (Aside from those with raft foundations/combined footings)?
Yes a portal frame with at least one fixed (moment bearing) joint on top can be stable.
Portal frames with pin joints at the base could theoretically bear moment, provided they have fixed joints on top of the columns, but they are not advisable.
In fixed base and fixed top, a frame shares the moment at all joints and vibrates under lateral loads more in uniformity with the rest of the building. Also it lends itself to smaller cross sections, hence more ductile behavior and less stiffness. Too much stiffness in a portal frame invites all the shear concentrated over its length and is detrimental to the overall lateral performance of the building.