As said by Thomas in the comment, modern encryption algorithms should work for all kinds of data, regardless of whether the plaintext is partially known or not.
In both modes of operation (CTR and CBC), for stored data you should remember to get a new initialization vector and reencrypt your whole "message" (or database entry) when changing any part of the data, not try to reencrypt only the changed parts. (Though CBC is likely a bit more robust here – the attacker in the case of IV reuse only gets to see from which block on the data changed, while with CTR mode she can see the XOR difference between the old and new values, assuming she can observe both ciphertext.)
Is it possible for someone to discover the CTR counter value
knowing some of the bytes in the original message?
The counter value can be derived from the initialization vector, and thus is considered public knowledge (even if nothing is known about the plaintext). There is no weakness resulting from that, as long as you take care not to repeat your counter values.
With CTR mode, you don't get authentication, which means that an attacker who is able to change your message and observe your programs reactions to that might gain some knowledge about the contents. To avoid this, combine it with a MAC algorithm (preferably on the ciphertext), like a HMAC, which refuses decryption if failed. (Have an independen key for your MAC.)
The same is valid for CBC, just the methods of attacking are different. Here it is even more important to apply the MAC on the ciphertext, not on the plaintext, for the mentioned padding oracles.
As Ricky Dehmer mentioned in a comment, an attack scenario would be someone who got write access to your database, but doesn't have the key, and aims either to maliciously modify the data, or to use chosen-ciphertext attacks to break your encryption privacy.
Don't forget to include the initialization vector in the MAC-ed message.
For an encrypted database column, you could store the tuple (IV, Ciphertext, MAC tag) together, either in one column (most easily in this order) or in three columns, and for decryption pass them together to your decryption function, which should first check the MAC, and then decrypt the ciphertext.
A authenticated encryption mode like OCB or GCM already includes the authentication, so you have less hassle to worry about.