We generally recommend collecting at least a five digit postal code - this is the variable you have direct control over that helps a larger percentage of your transactions process at the lower qualified rate. Other information, such as the street address, can be helpful for other reasons, but does not effect your processing rate. Collecting the CVV is usually a good idea, as well, though again - this data does not effect your processing fees.
It's been our experience that for SaaS providers such as yourself, traditional fraud is lower than you would see for merchants shipping physical goods. You'll still want to protect against chargebacks, however, and there are a couple options outside of collecting a large amount of data from your customers.
You'll want to first be sure the descriptor that appears on your customer's statements is obviously tied to the product they've purchased. You'll also want to ensure that the number appearing on their statements is a number that they can call and quickly get information about what the charge relates to.
Once your chargeback rate gets much above 1% of total credit card volume, banks will begin to analyze your processing and conduct various risk reviews. However, unless you're experiencing an obscene chargeback rate out of the gate, the bank will usually work with you to lower this percentage before shutting you down completely. Should you start to see chargebacks approach that 1% figure, you can then look at additional options, like increasing the amount of information you're collecting.
Much of this is just good practice for any business - accurately describe your service, be responsive to customer issues, and provide a great product.