Well it kind of depends what you mean by "secure", but in general YES it is. Here are a few facts:
We don't use a HTTPS secure connection, but that is only used for shopping sites where you put in credit cards numbers.
Anyone who doesn't register at PokeBase (i.e. just visits the regular site) is completely anonymous.
For PokeBase, all things like passwords you use are encrypted in our database, so even if someone managed to hack into the site (very difficult) they still wouldn't see the passwords.
IP addresses and email addresses are only visible to me, the administrator. (This is the same as every other site.)
Everything you post on the site and put in your profile is public, except for your email address. So you shouldn't post your email address, postal address or other personal details. General location is fine, e.g. your country or state.
All the code behind the scenes on the site is written securely so it is extremely unlikely anyone can hack into the site and add viruses or anything.
So in conclusion, YES the site is secure.