Yes! Boomerang takes security very seriously. Boomerang verifies ownership of a wallet by having you sign a message. Your private key is never shared with Boomerang, because it only compares your signature against your public key.
Be aware that your wallet and corresponding Discord handle will be visible to all admins of the Discord server you’re setting the wallet for, because that's how they'll know which wallet to whitelist or send rewards to. In case of an unforeseen hack, this will be visible to malicious parties as well. That does *not* endanger your holdings, because, as mentioned before, Boomerang never knows your private key, and you never give Boomerang permission to spend funds. But, if there's a wallet with holdings you'd rather keep entirely private and unlinked to your Discord ID, you might want to consider using another wallet for Boomerang instead.