Check on Blockchain.info if blocks are currently full.
Currently (2017/7/12) many blocks are half full, so even with a very low fee you can expect to be included in one of these that are not full.
Around 10 sat/byte is sufficient. If your wallet lets you set this ratio manually, fine.
If you only can set the actual fee, keep in mind that a Counterparty transaction is usually around 280 bytes. The required fee is thus
0.00000010 * 280 = 0.00002800 or in mBTC that’s 0.028. In dollar terms it’s around $0.06.
Note 1: XCP send transactions carry another cost of (normally) 0.00005400. This is dust sent to the recipient along with the asset. In the above example the total cost is then 0.00008200 or $0.19. A new send type, which eliminates the need for this, will very soon be introduced.
Note 2: If your address has lots of dust, the size of the transaction may be much larger. In this case the fee will also have to be much larger. However, I wonder if 1 sat/byte will get propagated and confirmed too? If this is the case, a fee of 0.00002800 byte should always work … granted there still are some non-full blocks.