Nope. Never. PayPal uses the highest-level encryption to ensure all transactions are safe and secure. It also provides both buyer and seller protection to safeguard the transaction and will reimburse you should things not go according to plan.
PayPal monitors every transaction 24/7 to prevent any fraud, phishing or identity theft and when you use PayPal your financial information is never visible to the online retailer.
See PayPal’s security page for more info on exactly what you’re covered against. Once you’ve set up your Paypal account with your email and password, though, it’s up to you to keep those safe from prying eyes.