This post is based on a visit in April 2021 when not all facilities were open due to Gov restrictions. Please check their website for up-to-date information before visiting or making plans.
At time of publishing, Durham Botanic Garden is free to visit and open Monday-Friday 10am - 2:30pm. You do not need to pre-book and can just walk in. The cafe, toilets and indoor facilities/bird hides are currently closed. I would still recommend as a nice peaceful walk with points of interest or a place to meet up with friends/family (within Gov guidelines) and it is lovely for a stroll. There is plenty of seating throughout the gardens and it wasn't too busy at all on the day of our visit in the school holidays.