Answer
$27$ cards
Work Step by Step
There are $26$ of each red and black cards in the deck. In the worst case scenario, you pick all the black cards first, meaning the first $26$ cards are black. This means the $27$th card is red. The number cannot be any higher than this, as there are 26 cards, so this is the maximal number of red cards drawn, so $27$ is the answer.