2 Timothy 3:16 (KJV)

All scripture is given by inspiration of God, and is profitable for doctrine, for reproof, for correction, for instruction in righteousness:

Celebrating birthdays isn't a pagan ritual itself, the issue comes in how you choose to celebrate them. In the bible kings celebrated their births. This is why Jesus was given gifts to celebrate his birth from the wise men, because he is the king of kings. The bible says that through Jesus we are all made into kings and priests. So celebrating our birth is also fine. 

Now here is where the problems come in. When we celebrate our births or our loved ones births, we need to be sure to give God all the praise and glory for the gift of that person in our lives, or for the gift of life he has given us. The pagan part of celebrating Birthdays comes from a few simple things. The first is to turn the birthday person into an idol for that day. In old pagan traditions the persons birthday was when they were celebrated as a God. "When a pharaoh was crowned in ancient Egypt, it was believed that he or she was transformed or reborn into a god." By singing the "birthday song" and making that person the very center of attention for that day, they are being made into an idol of worship.  I want to be clear on what I mean here, The "Happy Birthday" song itself isn't pagan and doesn't have pagan origins. It is a relatively modern creation that began in the late 19th to early 20th centuries. However, the broader tradition of celebrating birthdays by singing songs or worshiping the birthday person has connections to various ancient practices and has pagan roots.To avoid this, make sure that praising God for the person and thanking Him is at the center of the day. 

The second pagan practice of birthdays comes from the Birthday cake with the child's name written on it decorated with candles. Biblically speaking, names have power. That is why often times in the bible God gave his people different names. It's also by the name of Jesus we are saved. When you place your child's name on a cake surrounded by fire, this is a modern version of having your child "go though  the fire", which is a pagan ritual of child sacrifice to false gods.  God says he hates and doesn't want his children participating in this pagan practice. "Deuteronomy 12:31 Thou shalt not do so unto the Lord thy God: for every abomination to the Lord, which he hateth, have they done unto their gods; for even their sons and their daughters they have burnt in the fire to their gods." So if you write your child's name on their birthday cake make sure not to place burning candles around it, or just avoid birthday candles all together. 

The candles on a cake for a birthday are also another pagan ritual to worship pagan Gods. "The candles came first. In ancient Greece, round cakes were baked in honor of Artemis, the moon goddess. Candles were placed on top to represent the glow of the moon. In Greece, as in many ancient societies, it was believed that the smoke of the candles would carry their prayers to heaven (in this case, to the moon)." WordPress.com.  "As for blowing out the candles, the Greeks believed rising smoke took one’s prayers and wishes up to the sky, where the gods lived. In Europe, birthday candles were often kept burning all day both to celebrate and ward off evil spirits." TheFarmersAlmanac.com. As you can see burning candles, specifically on a birthday cake, is a pagan tradition that offers prayers to false Gods. You can easily avoid this pagan ritual by simply not putting candles on the cake. 

So overall it's okay to celebrate your and your loved ones birthday. Just make sure that you avoid all the pagan rituals that often come with it. Most importantly, remember to give God the glory and the honor and thanks for the life he has gifted you with.