If you are already settled on using DropBox, then your only choice at the moment is to use a 3rd party program such as PGP (or free/open GPG) to first encrypt the file and place it into your drop box. I know you said you wanted to use public/private keys, but as an alternative, you could also use 7-zip to create secure archives with AES encryption based on a pre-shared key (password).
If you have not yet decided on DropBox, you can look at similar online storage systems such as SpiderOak, Cryptoheaven, Mozy, etc. which perform client-side encryption of the files prior to uploading them into the cloud. You control the encryption key, and therefore the storage providers cannot access the data. I am not sure whether you can import existing private or public keys into those solutions or not, however.