There are many kinds of curses, and Christians are certainly affected by some.
The first curse against any humans is found in Genesis 3:17:
To Adam he said, “Because you listened to your wife and ate fruit from the tree about which I commanded you, ‘You must not eat from it,’
“Cursed is the ground because of you;
through painful toil you will eat food from it
all the days of your life.
Clearly all humans, including Christians, live under this curse.
Another type of curse is the so-called "Generational Curse:"
Our ancestors sinned and are no more,
and we bear their punishment. Lamentations 5:7 (NIV)
I think it's clear that this affects Christians, too. Not only in the way scripture talks about these generational curses, but in every-day life. Drug-addicted parents tend to have children with drug addictions--or other related social problems. And the cycle is very hard to break!
I think the question you're really getting at is, are curses "cast" by one person on another, effectual against Christians.
Proverbs 26:2 says:
Like a fluttering sparrow or a darting swallow,
an undeserved curse does not come to rest.
Proverbs is not always considered to be "hard and fast truths," though, and more "words to live by," so this may not mean that "undeserved curses" are never effectual. It's also unclear (at least to me) what type of curse this is actually referring to.
This article contends that Christians can be affected by curses (or other demonic/occult practices, such as fortune telling), especially when they open themselves up to such influences. But that through Christ, we can always overcome these attacks and negative influences.
As a personal side note, for which I have no references yet (I'll try to add some later):
Most Christians believe in spiritual warfare in one form or another. If spiritual warfare is a reality, then surely calling upon demons to harm Christians would be an act of spiritual warfare. And while I believe Christ can always overcome these attacks, I don't believe that always means Christians will never experience the effects of these attacks.