Because reflexes are automatically programmed, they do not rely on conscious or unconscious brain activity.
The interaction between reflexive and conscious responses can be more complex.
In certain situations, a brain-moderated response would be too slow, so reflexes come into play. Reflexes are automatic, hot-wired responses to particular, extreme stimuli. For example, if you pick up a very hot plate, you will probably instantly drop it. But if you pick up a hot plate that is valuable, you might reflexively drop it and, a split second later, consciously catch it. The two responses might vie with each other for a second or two as you juggle the plate while searching for a surface to place it on.
As mentioned in the above answer, you may learn to quickly react to certain stimuli, but if you generally want to get better reaction time then I recommend working on your agility(mental and physical) for reaction time.
You can start by
Improving physical agility
Suicide runs are a great way to improve your speed, and ability to
switch direction quickly and effectively. 
Do ladder drills.
The main objective of agility ladder programs is to promote a wide
range of different foot and movement patterns. Through practice these
movements will become second nature and the body will be able to
respond quickly to various sport specific movement patterns.
Improving mental agility
- Eat the right breakfast foods. Hard-boiled eggs (vitamin B) to increase verbal and visual performance. Bran cereal (rich in zinc) because Zinc plays a pivotal role in cognitive stability and memory formation. It also improves your skin tone. Antioxidant rich fruits and vegetables, they help with the increase of mental capacity and memory. Small cup of coffee (caffeinated tea) first thing in the morning to improve your mental performance and memory as well as increasing your concentration
Eating the right kind of breakfast is one of the primary ways you can
boost your focus at the beginning of the day. Dr. Mehmet Oz from The
Dr. Oz Show told his audience it is important to consume three foods
during breakfast that have been linked to boost your concentration to
prevent short term memory. 
- Exercise during the day.A short workout at any point during the day to increase mental performance. It also helps with mental health and agility by reducing stress, boosting mood improving chemicals in your brain, relieving anxiety, increase relaxations, and increasing creativity
Working out can have positive effects far beyond the gym (and beach
season). Gaining self-confidence, getting out of a funk, and even
thinking smarter are some of the motivations to take time for exercise
on a regular basis. 
- Read more. Reading engages many parts of your brain linked with memory, cognition, and imagination. Reading also improves mood and increases relaxation.
Reading is a great activity because it can stoke the imagination and
ignite so many different parts of the brain. 
- Play games (with friends). Choose those games that require skill and multiple levels of cognition to help improve focus and memory retention. Play the game a few days a week to engage your mind and increase your cognition.
Games are a great way to build up your brain muscle. Even fast-paced
action video games may boost your ability to learn new tasks,
according to a study in the journal Current Biology. The study found
tentative evidence that video games may increase your attention span,
reaction time, and task-switching ability. 
- Learn something new. Learn a new way to complete your normal routine and your everyday tasks. These tasks help your brain create new neural pathways.
According to the Alzheimer’s Association, research shows that keeping
your brain active increases its vitality. Doing new things in new ways
appears to help retain brain cells and connections. It may even
produce new brain cells. In essence, breaking out of your routine can
help keep your brain stay healthy.