Many former Picnik users have this question. Unlike Picnik, when you open a photo to edit in PicMonkey, you aren't uploading your photo to the site. All the editing takes place locally on your computer. So if you have lots of photos to edit, you can just open each one as you go.
Here's a hint: If you're using a browser that supports Drag & Drop (Chrome or Firefox), open the folder containing your "to be edited" photos on your desktop. Then, as you edit and save each photo, you can quickly drag the next one onto the PicMonkey page to open it. You may find that it's just as quick or quicker than opening a file from Picnik's photo basket used to be.