You can prepare Dispel Magic to instantly dispel whatever spell a spellcaster casts. However, this will not "counter" their spell, rather, it will dispel it as soon as they cast it. So any spell with a duration of Instantaneous will be unaffected. This will only end the ongoing effects of spells which create ongoing effects.
This is not at all the same as "turning Dispel Magic into a spell with a casting time of Reaction". From the Player's Basic Rules, page 72:
When you ready a spell, you cast it as normal but hold its energy, which you release with your reaction when the trigger occurs. To be readied, a spell must have a casting time of 1 action, and holding onto the spell’s magic requires concentration (explained in chapter 10). If your concentration is broken, the spell dissipates without taking effect.
When you Ready a spell, you cast it with your action, then hold on to the cast spell until your reaction is triggered. You burn the spell slot when you ready the spell, and you have to maintain concentration until your reaction is triggered, or the spell will be wasted.
This is an extremely strange way to try to go about this, because there is a spell called Counterspell (Player's Basic Rules, page 86) which does exactly what you're looking for. It nullifies spells, including Instantaneous ones, it has a casting time of Reaction, and you choose which spell slot you cast it out of when you cast it. It's the same level as Dispel Magic and does everything you want.