The Bible is silent about holy water the way it is used today. For baptism, Matthew 3:11 speaks of "baptizing with water for repentance," with nothing in the context suggesting that the water itself is holy. Baptism is a symbolic ritual, identifying oneself with Christ’s death, burial, and resurrection. A closer parallel to the modern use of holy water would be God’s laws in the Old Testament, which required cleaning ritually unclean things with water to purify them before one could touch them (see Leviticus 15, 16, and 17:15). See also a specific reference to ritually unclean people in Numbers 19:17.
Holy water is now permanently retained at the entrance of Catholic churches, blessed at the first of each lunar month, and sprinkled over patrons as they enter. This practice was created to supplant the pagan celebration of the new moon, according to Canon 65 of the Council of Constantinople (691). According to the Catholic Encyclopedia, the earliest modern uses of holy water appear in the ninth century. With that, coupled with the New Testament’s silence regarding the practice and use of holy water, it can be concluded that the tradition of holy water was created for the sole purpose of putting a pagan ceremony out of commission, using a scant few biblical references to water for purification.
Any practice that makes us feel closer to God and furthers our walk with Him should be encouraged (cf. Romans 14, esp. v23). But also consider 1 Corinthians 6:12. If a practice is beneficial to a relationship with God, keep it; otherwise, throw it away. This is all the more true when said practice has little biblical foundation. The Bible nowhere instructs Christians to use "holy water" in any way, shape, or form. The Catholic use of holy water is not biblical.