just like any other online transaction, FULL details are required. However, majority are already quite secure (with verisign and etc certified). Not easy to get these certification.
Same applies when you buy airline tickets online.... it's a matter of being comfortable online.
Yes, there are precautions. Careful of scam sites (eg the m2u scam sites), keyboard listeners etc. Use a proper antivirus on your computer and using a CC to pay will be better because you can dispute if there was an illegal transaction. Make sure you use a strong password too.
I've been doing online transactions since early 90s so I'm very comfortable with online purchasing. Never had any issue with any payment.
i also recently used myEG. Excellent service. delivered my roadtax within 24 hours to my doorstep.