The clash, which was played at Brighton Women's home ground - Crawley Town's Broadfield Stadium - was paused for 14 minutes while the emergency was being attended to.
The players waited on the pitch initially, but soon went to the dressing rooms.
The incident happened with just three minutes of the clash remaining.
Medics came to tend to the supporter in the stands.
The players then re-entered the pitch to finish the game, seeing out a 1-1 draw.
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Spurs posted on X: "Play is back underway at Broadfield Stadium following a short break due to a medical emergency in the crowd.
"Our thoughts are with the individual receiving care."
Brighton added: "Everyone at Albion sends their wishes."
Tottenham had broken the deadlock in the 75th minute through Bethany England.
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But Brighton's Marisa Olislagers bagged an equaliser just seven minutes later.
Neither side could find a winner after the restart and the 14 minutes extra time.
Brighton remain in fifth after Saturday's tie, with Tottenham staying six behind points them in sixth.
A further update on the condition of the supporter in question is yet to be released.
WSL clubs will enter the winter break after this weekend.
The pause takes place between December 16 and January 18, with the WSL season ending on the weekend of May 10 and May 11, 2025.