@Mike and Ninefingers both give good answers.
Would using a login prompt really be enough security for me have passwords and maybe even banking information up there?
No. Let me put it this way. Would you use an ATM machine if you noticed several people sitting in cars with video recorders trained on the PIN pad? Thats what you are doing when you use just a username and password to allow access to your banking information. Many people are watching the data being sent from the server to you and back again. It is the same with passwords. Any information being sent between a server and you without encryption is likely being watched by someone.
I know there is the obvious brute force which is the caveat to many login systems, but how reliable could mine be?
Your system could be very secure if you were willing to spend on the order of $10,000 (10^4 for localization), plus ongoing operating costs. However, I'm guessing you were looking to do this for free, or maybe a small monthly sum. In that case I would hesitate to store anything more sensative that your grocery list on your wiki.
If that wouldn't be enough, is there some way of encrypting my website?
Yes there are several posibilites for encryption, but as @Mike points out, if you want the remote server to do more than just hold a file for you, you need a significant amount of infrastructure. Back to the ATM machine... Adding encryption would be like putting a privacy screen between you and the suspicious people with recorders. That will take care of someone seeing your PIN number, but of the ATM machine is encased in stiff cardboard, as soon as you leave someone will rip open the ATM and take out all the cash.
The internet is convenient, but it is a threat rich environment. Instead, buy a USB flash drive, install portable encryption software and portable wiki software, and back it up regularly.