It all really depends on the account.
Are most of the weekend requests one-off jobs (move in/out, Airbnb, emergency)? If so, then I would definitely charge a weekend rate. Make sure it covers your increased payroll expenses. I would also ask your current staff if they are interested in putting in the extra time. More money on staff paycheques is great, but you don't want to risk burning out your team.
Or, are these jobs reoccurring? If so, then a weekend crew might be necessary. Maybe offset a crew to work Wed-Sun. I always find it is better to have full-time employees who have full-time expenses (rent/mortgage, bills, kids) with a few part-time backups. If you only staff your weekends with part-time staff, you'll run into them calling off regularly.
Having staff work on the weekends also opens the door to staff needing office support. Will someone be in the office to assist with questions/concerns/pricing/etc?