The antenna cable does not carry any voltage, but if you remove the antenna to make sure you have a good ground, then disconnecting the battery might be a good idea. If the repairs are done right, your reception should be greatly improved unless something is wrong with the radio.
The ideal is to have an unbroken cable from the antenna to the receiver. Any splices or connectors along the way are going to degrade the signal. If you have to use them along the way, then use the best connectors or splices that you can get and follow instructions to the letter, this will minimize your losses.
In most cases, the only losses you will notice will be on the fringe stations. Local stations are usually not affected by some small losses.