For the most part, the Crimean War had very little impact on the United States. However, some lessons from the Crimean War were learned by American military men and were applied to the Civil War. Other lessons that perhaps should have been learned were not. So the real impact of this war on the US was in the impact it had on the Civil War.
The US government sent the Delafield Commission to try to learn lessons from the Crimean War. It took from the war the lesson that a modern army needed to be professional and fight in a very methodical way. It learned the importance of engineering and artillery. However, it managed to miss the importance of rifles. It also brought back new ideas about cavalry organization and tactics. Finally, the Crimean War and Florence Nightingale had an impact on the US in that Nightingale's efforts led to efforts by the US Sanitary Commission to provide healthier conditions for soldiers in the Civil War.