Not so sure... That map looks like it would seriously disrupt the oceanic currents, and thus, the weather patterns. Not to mention, all of the oil producing countries are under water (sudden spike in oil prices).
Hmm, let's see:
Now, if most of northern Africa and Indonesia sank below see level (how deep?) AND the sea levels remained unchanged, the only obstruction to the Northern Equatorial current would be the American landmass. Otherwise the water would circulate much more freely. I don't think America or Europe would experience major change of climate. Southern Europe might experience seasonal hurricanes, the west coast of India would become drier (I think) and the Monsoons would be affected, though I don't know how exactly.
On the other hand, look at the benefits:
- Russia would finally get what it always wanted: a warm southern oceanic port

- if the land didn't sank very deep, a huge area of shallow seas would be created, which would be a blessing for the marine life. Eventually, corals would cover the bottom of the Saharan Sea

- Israel would finally be safe (though obtaining fresh water would become very difficult).
- the main shipping bottleneck (Strait of Malacca) would be gone.
- DR Congo would gain much better access to the sea.
- Trade between Europe and Asia would get much easier.
- India would lose its two unstable Muslim neighbour states, the constant source of tension in the region.
- South Korea would become an island, therefore it wouldn't need American military presence any more.
- Lake Zimbabwe would become a source of fresh water for the entire southern tip of Africa

- The Aral and Caspian seas would become true seas again

- Earth's population would decrease by about 1.5 billion, which would reduce our environmental footprint.
- The main source of religious conflict in the world would all but disappear.
Basically, we'd trade vast areas of desert for shallow seas full of life. Not a bad choice.