No.
At the energies involved, no known net material could stop an asteroid which was big enough to be a threat. Even if it could, "stopping" the asteroid still leaves you with all that energy to dissipate. And it will dissipate in a massive explosion.
The energy
You can get a rough idea of the scale of these sorts of problems by calculating the energies involved.
Using an impact calculator, we find a rocky 10km asteroid (a once in 100 million years size) with a density of about 3,000 kg/m^3 has a mass of about 1.6e15 kg. At 18 km/s that packs a wholloping 2.5e23 J.
That is roughly the energy if we put all our Uranium-238 in a reactor at once. That's 10 times all our coal reserves going off at once. It's 60,000,000 megatons, 1 million Tsar Bombs. If it hits land, it will produce a fireball over 100 km in diameter and a crater 75 km in diameter and 1km deep.

No net of any known or speculated material can stop this. It will pass through it like it wasn't even there. If the net somehow held, whatever is holding it would break. See World Outsider's answer for details.
What if the net held?
Let's assume the net and whatever sky hooks it's attached to held... somehow.
Most asteroids are piles of gravel loosely held together by gravity. The asteroid has so much mass and so much energy it will disintegrate and pass right through the net. A small amount of energy will be lost passing through the net, causing an explosion, but most will be retained. Now instead of one big impactor, you have a shotgun of smaller ones, still with the same total energy.
What if the net held and so did the asteroid?
Let's assume the net holds, the sky hooks hold, and the asteroid does not disintegrate. That isn't much use unless we can also stop it. As they say, it's not the fall that kills you but the sudden stop at the end.
When 1.6e15 kg goes from 18 km/s to 0, the energy has to go somewhere. Instead of impacting the ground and exploding, your asteroid impacts the net and explodes. How much damage it does depends on how fast it explodes and how high up the net is. The higher the net, the slower it can decelerate the asteroid. We need to make this stop as unsudden as possible, or well outside the Earth's atmosphere where it will do less damage.
Of course, in order to stop it we need 2.5e23 J of energy... handwave.
To calculate how fast the net has to decelerate the asteroid, we use the kinematic equation...
$$v_f^2 = v_i^2 + 2ad$$
- $v_f$ is our final velocity, 0
- $v_i$ is our initial velocity, 18 km/s
- $a$ is how fast we have to decelerate it.
- $d$ is the distance we have to decelerate it.
We want to know for a given height above the atmosphere, d
, how fast do we need to decelerate the asteroid, a
? So solve for a
and plug in various values of d
...
$$v_f^2 = v_i^2 + 2ad$$
$$v_f^2 - v_i^2 = 2ad$$
$$\frac{v_f^2 - v_i^2}{2d} = a$$
Plug in our constants $v_f$ and $v_i$...
$$\frac{0\frac{km^2}{s^2} - 324\frac{km^2}{s^2}}{2d} = a$$
$$\frac{-162\frac{km^2}{s^2}}{d} = a$$
And let's play with some distances.
- Maybe your net is hooked between two very tall mountains, 9 km high. It needs to decelerate at 18 km/s^2, or 1800 g's, in one second. 2.5e23 J being released in one second is 2.5e23 Watts or a million times the energy received from the Sun in that time. It explodes.
- At 100 km, where space begins, and hung from... something... it needs to decelerate at 1.62 km/s^2 or about 165 g's over 11 seconds releasing 2.3e22 Watts, or 100,000 times the energy from the Sun, into the atmosphere.
- At geostationary orbit 42,000 km out, it only needs a leisurely 3.8 m/s^2 for 4700 seconds. This is 5.3e19 Watts or 300 times the power received from the Sun for over an hour. It will be very hot, I'm not sure how to calculate how hot, but you probably don't want to let it get too close to the Earth.
If you'd like to know how much damage it will do when "it explodes", I found this delightful paper for you: "Simulation-based height of burst map for asteroid airburst damage prediction".
Play the long game.
If you have enough lead time to build a net, there are much better options. You take advantage that space is really, really big and the Earth is relatively small and it's moving very, very fast. It's a 13,000 km ball moving at 30 km/s in a volume space of millions of millions of millions of km^3. And it's moving at 30 km/s.
If you give the asteroid a small but constant nudge early enough it will miss.
One Falcon Heavy rocket produces about 25,000 kN of thrust. If it pushes our 1.6e15 kg asteroid it will produce an acceleration of 1.5e-11 km/s^2. That's not a lot, but over a year (I don't ask you about your sky-hooks, you don't ask me how we fuel a rocket on an asteroid for a year) it's a 0.5 m/s change. Again, not a lot, but over a year that's a 16,000 km difference which is enough to miss the Earth.