What serious problems did they fear? It does not seem they feared it too much since they gave it additional powers compared to the House to confirm judicial appointments and ratify treaties (a power they originally wanted to give entirely to the Senate).
From the Senate History:
"The framers of the Constitution created the United States Senate to protect the rights of individual states and safeguard minority opinion in a system of government designed to give greater power to the national government. They modeled the Senate on governors' councils of the colonial era and on the state senates that had evolved since independence. The framers intended the Senate to be an independent body of responsible citizens who would share power with the president and the House of Representatives. James Madison, paraphrasing Edmund Randolph, explained in his notes that the Senate's role was "first to protect the people against their rulers [and] secondly to protect the people against the transient impressions into which they themselves might be led."
"In the early weeks of the Constitutional Convention, the participants had tentatively decided to give the Senate sole power to make treaties and to appoint federal judges and ambassadors. As the convention drew to a close, however, they moved to divide these powers between the Senate and the president, following Gouverneur Morris' reasoning that "As the president was to nominate, there would be responsibility, and as the Senate was to concur, there would be security." Due to the concern of individual states that other states might combine against them, by a simple majority vote, for commercial or economic gain, approval of a treaty would require a two-thirds vote. In dealing with nominations, the framers believed that senators -- as statewide officials -- would be uniquely qualified to identify suitable candidates for federal judicial posts and would confirm them along with cabinet secretaries and other key federal officials, by a simple majority vote."
https://www.senate.gov/artandhistory/history/common/briefing/Origins_Development.htm