Can I safely connect two laptops to get pictures from one to the other if one of them is not working?

It depends on how the computer has failed: If it is a software issue, you can try a bootable USB or CD. Files on the HDD should be readable and the PC can be connected. In the case of a bad screen, an external monitor may be used temporarily. A bad RAM stick can be removed if you have more than one and a required amount of ram on the other sticks. A bad CPU or motherboard could prevent the computer from being usable at all but the hard drive can be extracted and a cheap usb adapter connected to the working computer. If the hard drive has failed it may be possible to use a data recovery service but this option can be very costly.