The short answer is YES, PayPal is secure. There have been many false rumors about PayPal security, these rumors are based on incidents with people who have been victims of Phishing scams. This happens with all major online companies who take money over the internet (Google, Ebay, etc.). Criminals setup fake websites that look just like PayPal and others, then send out emails that ask you to go to a link in the email and update your personal information. If you you follow that link and enter your user name and password you go to an official looking website and they ask for your personal information like credit card numbers, etc. Legitimate companies never ask you to do this! There are two things you can do to avoid this.
1) Make sure the lock symbol is showing on your browser which means you have a secure connection. If the address bar is not green and there is not a lock symbol next to the address, you are not on a secure site.
2) If you are not sure whether the e-mail you received is legitimate, provide a bogus user name and password when asked. If it is accepted then they don't really know who you are and it is a Phishing scam.