Ideally, a bathroom for an elderly person should be on the first floor with a large, walk-in shower outfitted for handicapped access. If you can't afford an entirely new shower, however, you can do a few things to make your existing shower safe. Installing a shower chair, accessibility features and safety alert features will greatly enhance the safety of your shower.
Put a rubber mat in the bottom of the shower. Many varieties of mats are available, including those with suction bottoms for stability. The more textured the mat, the more shower safety it will provide.
Install a grab bar at arm level on the side of the tub, perpendicular to the edge of the tub. You can also install a suction grab bar directly on the edge of the tub. This will give an elderly person something to grasp for stability when they're getting in the tub because the process of lifting the leg over the tub edge can lead to dangerous falls.
Install another grab bar at hip or waist height on the wall or backing above the tub, parallel to the edge of the tub. This adds to shower safety in two ways: if an elderly person falls in the shower, they can use it to pull themselves up. Alternately, if you wind up installing a shower seat, this bar will help an elderly person pull up from sitting. You may also want to install a diagonal grab bar, with the highest point being at the front of the shower.
Install a wireless doorbell just outside the tub, toward the lip of the tub, if the elderly person has a roommate or spouse at home. Put the receiver/bell part in the living room or a central part of the house. This allows the elderly person to ring for help if they fall.
Keep a cordless phone in the bathroom, if the elderly person lives alone. You can buy phones that only require chargers, and have a base set that plugs in elsewhere. Keep it near the shower. Alternately, have the elderly person take a cell phone into the bathroom and keep it near the shower. This adds to shower safety by giving the elderly person a way to call for help should they fall.
Install a shower seat if the elderly person has trouble standing for the duration of the shower. The shower seat has holes in it for water flow and handles on the side for easy sitting and standing. The shower seat sticks to the bottom of the tub, and provides safety and stability for the elderly.