This is a complex question, so let's break it down and answer it in parts.
Were animals wild before the Fall?
Animals were certainly "wild" before the fall, in the sense that they weren't domesticated. In Gen. 1:24, one of the categories of animals created is translated "wild animals" in NIV; the Hebrew word here (hay) is similar to the English word "beast" in that, depending on the context, it can either refer to animals in general or more specifically to dangerous/unruly wild animals. In this context, since it's a separate category from the "livestock" and "creeping things", the latter meaning should be understood. Domesticating the natural world was a task that God gave to Adam and Eve in Genesis 1:26 & 28.
Were wild animals dangerous to humans before the Fall?
From Romans 5 (in particular, verse 12) it is plain that people are subjected to death as a result of Adam's sin, and thus we cannot believe that animals might have killed people before the Fall. While there were wild beasts already in the world, they were not dangerous to man yet. Mankind's strife with nature was a result of the curse God laid on us after Adam's sin (Gen. 3:17-19).
Did the Fall affect the natural world?
Yes, see Romans 8:18-25. The natural world is subjected to futility as a result of Adam's sin and God's curse. This futility is a major theme in the book of Ecclesiastes, see especially 1:5-9. Things happen all the time in nature, but with no apparent goal or benefit. As Paul explains in Romans 8, nature is groaning and eagerly awaiting the return of Christ, even as we ourselves are.
Did carnivory begin at the Fall?
There are many Christians who believe that it did, but that inference cannot be made from Genesis alone. One thing that is plain is that humans did not begin to eat meat as a result of the Fall. That's a common misconception, but reading the Scriptures carefully shows that actually the eating of meat was a gift from God, given after the Flood, over 1500 years after the Fall:
Then God blessed Noah and his sons, saying to them, “Be fruitful and increase in number and fill the earth. The fear and dread of you will fall on all the beasts of the earth, and on all the birds in the sky, on every creature that moves along the ground, and on all the fish in the sea; they are given into your hands. Everything that lives and moves about will be food for you. Just as I gave you the green plants, I now give you everything.”
~ Genesis 9:1-3
Eating meat is a blessing from God and not a curse from the Fall.
Genesis simply does not say when animals began to be carnivorous. Thus, Christians are divided over whether or not animals were carnivorous prior to the Fall. We must be somewhat cautious about assuming that our opinion of what is good for animals is the same as God's. As C.S. Lewis points out in Problem of Pain chapter 9, we do not know enough specifics about God's purpose in creating animals to say definitively whether or not it is "bad" for them to eat one another.
Therefore, those who believe that animals didn't eat one another before the Fall make more complex arguments to defend that opinion. They will point to some of the eschatological verses in Isaiah, namely 11:6-9 and 65:25, which show predator and prey animals living in harmony, and claim that these portray a restoration of the pre-Fall world. Therefore, there must have been no carnivorous animals before the Fall. (Irenaeus makes this argument in Against Heresies Book V.33, though his main argument here is eschatological, he clearly sees these verses from Isaiah as a literal restoration of pre-Fall creation.)
They will also argue that, because Romans 5 says that death entered the world through Adam, this should be taken to mean death in general (including that of animals) and therefore there cannot have been carnivory before the Fall.
On the other hand, Christians who believe that animals have been carnivorous from the beginning, will caution against inferring too much beyond what is said expressly by the Scriptures. Only human death is in the scope of Romans 5, and none of the Isaianic passages claim that a restoration of Eden is what is being portrayed.
When we observe the created world, we can see right away that every kind of animal has features designed to kill other animals, or to avoid being killed. Many plants also actively try to kill animals, and lots of organisms (especially fungi) are designed to eat the corpses of dead animals. Since animal death and carnivory seems presumed in the design of every organism, and Genesis 2:1-2 indicate that God finished creating everything by the 6th day (clearly pre-Fall), it would be natural to infer that animal death was always a part of creation. Augustine explains this in Book III.25 of On the Literal Interpretation of Genesis:
Someone is going to say: 'Then why do beasts injure one another, though they neither have any sins, so that this kind of thing could be called punishment, nor by such trials do they gain at all virtue?'
For the simple reason, of course, that some are the proper diet of others. Nor can we have any right to say, 'There shouldn't be some on which others feed.' All things, you see, as long as they continue to be, have their own proper measures, numbers, and destinies. So all things, properly considred, are worthy of acclaim; nor is it without some contribution in its own way to the temporal beauty of the world that they undergo change by passing from one thing into another. This may escape fools; those making progress have some glimmering of it; to the perfect it is as clear as daylight.
With the right perspective, we can see ecology itself as beautiful.
I myself would favor the latter view (i.e. that animal death was the original, pre-Fall design), but I respect those who think differently. It is not an easy question, and there are good arguments in both directions.
