Yes, people can be possessed by demons today. There is no particular Scripture to indicate that anything changed form the time that Jesus pronounced certain people as possessed to now.
It's very important to remember that Jesus is always trump. The disciples were amazed to find that even the spirit world (however that manifest itself) was subject to the the authority of Christ and his teachings1, not just when he gave them but when they went out to proclaim them2.
Also, it's not something focus on or draw our attention from the main message of the Gospel3. It's sort of an incidental problem that we overcome just like any other problem in this world. Undue focus on the realms of evil, spirit or physical, is unhealthy. In fact many things are often mistaken for spirit world problems that are just the fruits of our flesh. Sorcery is one example4.
Lastly, Christians cannot be possessed. The enemy has no authority over the temple of the Holy Spirit. Christians have been raised into life with Christ and are temples of the Holy Spirit. Those temples are are off limits to the enemy because the victory has already been won5 and we have been transferred from the kingdom of darkness to that of light6.
Mark 1:27 (ESV)
And they were all amazed, so that they questioned among themselves, saying, “What is this? A new teaching with authority! He commands even the unclean spirits, and they obey him.”
Luke 10:17-19 (ESV)
The seventy-two returned with joy, saying, “Lord, even the demons are subject to us in your name!” And he said to them, “I saw Satan fall like lightning from heaven. Behold, I have given you authority to tread on serpents and scorpions, and over all the power of the enemy, and nothing shall hurt you.
Luke 10:17-20 (ESV)
Nevertheless, do not rejoice in this, that the spirits are subject to you, but rejoice that your names are written in heaven.”
Galatians 5:19-21 (ESV)
Now the works of the flesh are evident: sexual immorality, impurity, sensuality, idolatry, sorcery, enmity, strife, jealousy, fits of anger, rivalries, dissensions, divisions, envy, drunkenness, orgies, and things like these. I warn you, as I warned you before, that those who do such things will not inherit the kingdom of God.
Colossians 2:15 (ESV)
He disarmed the rulers and authorities and put them to open shame, by triumphing over them in him.
Colossians 1:13 (ESV)
He has delivered us from the domain of darkness and transferred us to the kingdom of his beloved Son