The simple answer is very safe. Encryption technology converts your personal details into a secret code, which is then sent over the internet. It is a bit like someone giving you a locked box, and telling you that the key for the lock is somewhere in that drawer over there. Some companies, particularly on-line banks are using even more complex encryption technology.
Many people still insist on using passwords that are obvious or easily guessable.
Using your name, partner's name or date of birth etc is just not good enough. If a site asks you to create a password, a combination of numbers and letters difficult to guess would be the safest. Also, you can be aware of 'safe' sites. Those sites which are running a secure connection will force your web browser to display something along the lines of a locked padlock, or a whole key instead of a broken one. Highlighted are where and what you should look for.