The most important thing that you can do as a GSA as far as driving Red Cards is to keep a positive attitude and recognize cashiers that do a good job. I'm not just talking about squawking on the walkie "WAY TO GO X FOR GETTING US ANOTHER REDCARD!!" I mean actually congratulating the TM and make sure they know that they're doing a good job. When you are speed-weaving, make sure that you pay attention to which cashiers are consistently asking guests about the Red Card, and who isn't. It's been an issue at my store where a bunch of our cashiers will only ask when one of us is nearby. I don't mean that they don't ask every single guest, I'm talking about people who don't ask period. If you find that a certain cashier isn't asking, let your GSTL/ETL-GE know about it so that they can chose whether or not to follow up.
Nauz already touched on prizes, but my leadership as of late has been all about giving out monetary prizes since we've been so down on Red Cards. I don't think any stores in our district have been green since early October. Most weekends we are giving out some kind of gift card at the end of our GSA shift to whoever has the most Red Cards. For about 6 hours straight on BF we had power hours where whoever got the most Red Cards in an hour got a target swag item (Dog, lunch bag, or water bottle) as well as a $10 AMC giftcard. The effectiveness of this really depends on whether the majority of your team is intrinsically or extrinsically motivated.
One thing that I find helps, is that at the start of every week I'll print out the Cashier Report off of myReports and go over it with each cashier during their one-minute huddle. I'll identify where they're strong, and give suggestions on how to improve on their weaker areas. This also helped me keep them up to date on how they are doing and what they need to improve on.
Also, try not to let yourself get overwhelmed. I see a bunch of our newer GSA's quickly get overwhelmed by the constant barrage, and they lose their cool and it makes them slip up. As long as you're following policy, the lanes aren't on fire, and nobody is in non-compliance, you don't need to run around like a chicken with your head cut off. 90% of the time, the task can wait a few minutes. Adding on to that, guests always take priority. Yes, baskets/reshop/hangers need to be collected, but you can have a cashier do that or you can do it later. But if a guest/cashier needs you, that takes priority.
One last bit of advice, if you're helping a guest at the service desk/doing something, make sure that you are always glancing up every 30 seconds or so to make sure that the lines aren't getting too long. When I first started that would occasionally happen where I would be so immersed in helping a guest at the SD that I would fail to notice that suddenly we are backed up to softlines. That's something that took a while for me to get into the habit of doing.