Donald Trump wants the wall for one reason and it has nothing to do with illegal immigration, not one tiny bit. It has to do with his campaign promise and appeasing his base.
Trump's base liked him because he said he wasn't a politician, and they weren't happy with politicians because they hated President Obama. No, he wasn't a politician and had no clue at all how politics works. He was a businessman and a shady character to say the least, more like a NY racketeer. So when he came to Washington he found out that he's not the boss, he's not a king and certainly not able to do anything he wishes. He's 1/3rd of the government and has 1/3 of the power of government.
Nancy Pelosi is a politician and an expert strategist. She's been in politics all her life and spent the past 30 years in the 'establishment' of U.S. politics. For 31 years, Speaker Pelosi has represented San Francisco, California’s 12th District, in Congress. She has led House Democrats for 16 years and previously served as House Democratic Whip. In 2013, she was inducted into the National Women’s Hall of Fame at a ceremony in Seneca Falls, the birthplace of the American women’s rights movement. She's pragmatic, she's focused and she's determined. She's a force unlike anything Trump has ever met in his life, and it shows.