Does water freely absorb atmospheric CO2?
Er, the quick, simple technical answer is "Yes, water will absorb $\ce{CO2}$". However, there is a "but" coming. Actually, two "buts".

First, plain water will not be able to absorb much $\ce{CO2}$. Henry's Constant for $\ce{CO2}$ in $\ce{H2O}$ is $K_H=29.76 \mathrm{atm\over mol/L}$. Considering the partial pressure of $\ce{CO2}$ in air is roughly $3.5\times10^{-4}~\mathrm{atm}$, one liter of water can absorb, at best $1.18\times10^{-4}~\mathrm{mol}~\ce{CO_2}$ which is only about $0.1$% of a gram of $\ce{CO2}$. Not very much. (Note, I'm ignoring the conversion to carbonate, $\ce{CO3^{2-}}$, and bicarbonate, $\ce{HCO3^{-}}$, because in otherwise neutral water, they don't exist in appreciable quantities (see: Carbonic Acid)

Second, that's the thermodynamic best you can do. However, you have to contend with the kinetics and diffusion limitations as well. Basically, it takes time for the $\ce{CO2}$ to go from the air to the water, and then to get well mixed within the water. 

Now, there's a couple things you can do to help $\ce{CO2}$ absorption. First, instead of using neutral water, use and alkaline solution. For example, if you add $\ce{NaOH}$ to your water, the equilibrium will shift toward the carbonate ion, $\ce{CO3^{2-}}$:

$$\ce{2NaOH + CO2-&gt;Na2CO3 + H2O}$$ 

This allows orders of magnitude more $\ce{CO2}$ to be absorbed than would possible, basically, you become limited by the amount of sodium hydroxide you have and the kinetics. 

Speaking of kinetics, how can we help that out? Packed bed towers! Imagine having a tall column filled with beads. Blow your air up from the bottom, while allowing the alkaline solution to flow from the top. This will maximize contact between the two phases and enhance the transfer of $\ce{CO2}$ from the gas to liquid. 

And that is basically how scrubbers work in chemical plants, and one method of carbon sequestration. However, the very low concentration of $\ce{CO2}$ in air ($\sim\!400~\mathrm{ppm}$) makes pulling appreciable amounts of $\ce{CO2}$ from the air infeasible and an inefficient use of resources.