FIG. 1 is a top perspective view of the folding basket travel bed of the present invention showing my new design; FIG. 2 is a top plan view thereof; FIG. 3 is a right side elevational view thereof; FIG. 4 is a front elevational view thereof, the back elevational view being a minor image of the front elevational view; FIG. 5 is a bottom plan view thereof; and, FIG. 6 is a left side elevational view thereof.