Did the benefits of immigrating to the United States at the turn of the century outweigh the challenges?

To answer this question, one must weigh the benefits against the challenges, and discuss in detail what the experience of immigration tended to entail.
First, this raises the question: what factors led people to immigrate in the first place? In general, historians have accounted for several motivating factors. Many immigrants were impoverished, and came to the United States in the hope of finding jobs. Additionally, one should factor in those who were fleeing political unrest or suppression—it could be ethnically based, or religiously based, or it could be based in something like political turmoil, but a significant number of immigrants came to America in order to escape very real violence and oppression within their homelands. You should be aware that for most immigrants there would have been an expectation that their lives would be improved by moving to the United States, compared against the lives they had lived in the countries they had come from.
That being said, there were serious drawbacks as well. Upon reaching the United States, they had to cope with prejudice and anti-immigrant sentiment. In addition, they had to reckon with culture shock. In immigrating to the United States, they would be introduced to (from their perspective) often very alien customs and religious practices, language barriers, etc. Even the very experience of traveling to the United States could carry with it significant hardship. To reach the Americas from Western Europe could take roughly ten days (spent in often very poor conditions) while, to travel from Asia could take roughly twice as long. Furthermore, prospective immigrants did not have any guarantee of entry at the end of it. Inspectors had license to bar immigrants from entering the United States for any number of reasons, and there are many stories of families being separated in this way. Furthermore, consider this: for many immigrants, employment meant work in U.S. factories and industrial centers, in jobs which were miserable and exploitative, and life would be lived in tenements.
That being said, I'm hesitant to say that there is a universal answer to this question. The hardships were significant on both sides of the immigration process, and I think the most accurate answer to your question would probably be to apply it on a case-by-case basis. Individual experience tends to vary from person to person.