One way to get the convenience of a suspended ceiling in the basement without the look of a suspended ceiling is to expose your beams (the floor joists above) and put suspension ceiling panels between the beams.
Here are two alternatives:
1. IF you can move your electrical, plumbing etc. into the ceiling and leave the bottom two inches or so of the joists unobstructed then you can attach the metal suspension pieces to the sides of the joists and have an exposed beam suspension ceilng.
2. A better approach is to build 3-sided box beams from 1 X 2 wood and attach them to the bottom of the joists. The suspended ceiling panels will rest on top of the box beams. You will need metal cross bar pieces for every 4 feet of the suspended ceiling, depending on the ceiling panel size. You may have to sister lengths of 2 X 2 material to the bottom of the joists to get the clearance that you need for the electrical, plumbing, etc., and then attach or build the box beams over the 2 X 2.
You can use old wood, painted wood, stained hardwood or whatever you want for the exposed box beams. Make sure everything is level, which is easy to do with this system. (Check the existing floor/ceiling joists for level and make adjustments to this system to end up with a level ceiling.) You will have to cut the ceiling panels to fit between the joists. Save some extra ceiling panels as future replacements for panels that become damages or water-stained. Make sure you build in enough clearance to insert and remove the panels.