First of all, we are to avoid substances that control us. The only power we are supposed to be under the control of is the Holy Spirit. Eph. 5.18 says, Do not get drunk on wine, which leads to debauchery. Instead, be filled with the Spirit. Does drug use lead to excessive indulgence? Certainly. It takes control of people. For a believer, the Holy Spirit is to be your only controlling force. (There are other texts: 2 Tim. 4.5; 1 Pet. 5.8, and more.)
Now, in case there is an objection that medicines control us, we take two perspectives. The first is that the Bible allows the use of medicines, and the second is that the motive behind medicines is healing, not addiction and pollution.
Secondly, we are to avoid substances that pollute our bodies. Galatians 5.19 warns us to stay away from witchcraft, and the word for witchraft is pharmakeia. Ah, you may think this is just the paranoid church drowning witches in Salem on false accusations, but witchcraft in Pauls day was the art of poisoning. So, to me that question that follows is: Do drugs poison our bodies? Uh, yep. Talk to any addict. You can also read Proverbs 23.29-35 about the power that booze has to pollute our bodies, and the teaching is to stay away from such things that pollute our bodies. (Some use this point to argue that we should eat properly also. I agree.)
Next, the Bible teaches us to take care of our bodies. The Holy Spirit lives in us, and we are His temple, and so we are supposed to care for our bodies like sacred space (1 Cor. 3.16-19). Another place in 1 Corinthians (6.19-20) teaches us that we are supposed to our God with the way we treat our bodies. Though people claim that when they use drugs they see God (ha!), but I cant imagine anyone claiming that they use drugs as a way of honoring God.
My final point is that we are taught in Scripture to obey the laws of our society unless they go against Gods commands (Rom. 13.1-7; Dt. 17.2; Mt. 22.21; 23.2-3 and others). Since right now the use of illegal drugs is, well, illegal, duh, then a believer in Christ should no be a user.