There is no substitute for good reps staying down not just with the glove but also with the body. Teams I coach or help with get a steady diet of fielding grounders and doing pickups trying to keep the glove from leaving the ground even as they move laterally. This drill work increases range and allows a player to understand the concept of working up with the ball if need be as opposed to stabbing at the ball whether forehand or back hand. This drill work also leads to diving for the ball as they learn to move laterally from a lower position.
Players will ask, "In a game do you want me to go get the ball with my glove on the ground?" To which I reply, "In a game you react, in practice you work on keeping the glove down."
I am a big believer that one major reason players do not stay down is due to being out of shape to do so - meaning they have trouble remaining in a low position as they move. Their backs and legs are not used to it. Most have a habit of straightening up to move forward, back, left and right. And if the ball is hit right at them - this habit of straightening up allows the ball to travel one hop farther. And it is always the last hop that kills you
Here are a few clips of the pickups we routinely practice.
Pivot Back Hand This drill we focus on flipping the glove over and dropping the outside knee
Crossover Back Hand Again flipping over the glove and then after the cross over step dropping the outside knee
Forehand Pickups Keeping the glove down keeps the body down
Pickups Taking Back Angle This drill leads to diving as you roll the ball farther out
The key to these drills are rolling balls so the players can focus on form and plenty of reps. After a few practices they get stronger and keeping the glove down becomes more routine
Hope this helps.