Busted Crown is a Boss Relic that grants 1 extra Energy each turn, but restricts the options available when picking cards.
While the player has this Relic, they will be offered 2 fewer cards on Card Reward screens. Note that this doesn't just apply to post-battle rewards - Tiny House, Sensory Stone, and other opportunities to pick cards will also be reduced.
If your deck is at least nearly prepared to win the rest of the run, Busted Crown is effectively a free source of Energy. Early in the game, though, it can represent a crippling disadvantage.
When deciding whether to take the Crown, consider your options for getting more card selection. If you have Dream Catcher or a large amount of money, you may be able to make do. Bear in mind that the Rare boss reward cards will also be reduced!