Does Achebe show that politics debases the individual and the country in "Anthills of the Savannah"? How does he do so?

Certainly he does.  Sam, Ikem and Chris are all caught up in the mechanism of politics, and all of them lose at least a little of their integrity.  Beatrice, however, makes a conscious effort to remain removed from politics, and she is the one at the end who both survives and demonstrates integrity and hope for the future.  Achebe uses the contrast between the characters to demonstrate the negative influence of politics.

Sam is clearly the most debased, which makes sense, as he is the most political.  He is so caught up in the politics of his position that he has failed to think rationally about the issues facing him.  He is unconcerned with the fact that many of his countrymen are suffering from famine and drought - he cares only about the appearance of protesters in the city.  The ruthless murders he orders, particularly of Ikem, demonstrates the loss of his individual and human concerns.

Ikem, though more honorable than Sam, gets caught up in politics, too.  He fails to see his countrymen as individuals, lumping them together and assuming that he knows what is best for them.  His editorials and his protests do focus on the needs of the common people, but they criticize on philosophical grounds and do not seek to find reasonable solutions that are workable to end immediate suffering.  In this way, Ikem has lost touch with common people he fights for, as demonstrated by his interaction with the taxi drivers.

Chris is the most balanced of the three, but his attempt to maintain both his friendships and his politics lead to ... not being able to maintain anything.  He knows that Sam is a tyrant, and yet he is unwilling to give up his position in the Cabinet.  He foolishly believes that he will be able to fix the corrupt system from the inside, not understanding that by staying there he is condoning the system.  His death at the hands of the corrupt public is symbolic of the death of his integrity at the hands of a corrupt friend and boss.

Beatrice, in contrast, maintains true to herself and to her country.  She does not go to extremes - she sees that her country has problems.  She is willing to speak about those problems openly, but she does not rush headlong into "fix it" mode, as Ikem does.  Nor does she try to prop up the corrupt system, as Chris does.  She works and lives each day by itself, allowing her individual daily activities to speak for themselves.  In this way, she not only survives, but helps to give hope for the future in the naming ceremony that ends the novel.