The Swiss champions will run out at St Jakob-Park on matchday two having failed to win any of their last four matches in continental competition.
Basel's European adventure started in positive fashion as they recorded a 5-0 aggregate win over Estonian minnows Flora in the two-legged Champions League second qualifying round.
A 1-0 away win over Molde followed in the next phase before a 1-1 draw in the return fixture.
FC Cluj of Romania awaited in the play-off and proved too good for Vogel's men, first coming from behind to take a 2-1 away win, then ensuring their place in the top competition with another victory on home soil.
That saw Basel relegated to the Europa League and they opened their Group G account with a tame 0-0 draw at Sporting last month.
The wait for a European win is now long overdue and although they head into the fixture unbeaten in six outings across all competitions, they have drawn four of those games and will be looking for a greater cutting edge against Mario Been's team.
Been's Genk side will provide confident opposition after extending their undefeated streak to eight matches on Saturday.
They have won four matches in succession and have found the target in each of their 15 competitive matches this term.
Genk appeared in last season's Champions League group phase but are making their first appearance in this stage of the Europa League.
They were the only side to win last time out, defeating Videoton 3-0 to take an early lead in the standings.