If the police had questions about events leading to the crime they would just go door to door and collect information. If police had questions about the crime that weren't found on site they would just go door to door and collect information. If police had questions about stuff that happened after the crime they would go door to door and collect information. Associations can be made from a lot of little observations from many eyes.
Newspapers (which were reliable in those days). The radio; wasn't it about this time that Crippen was caught by the first transatlantic radio message? In the UK the police called for help with the use of whistles or just shouting.
@Thinkerbell Most people in the USA didn't have phones then.
"‘By 1948, the 30 millionth phone was connected in the United States; by the 1960s, there were more than 80 million phone hookups in the U.S. and 160 million in the world; by 1980, there were more than 175 million telephone subscriber lines in the U.S.’"
That's true, most people didn't have phones. That was one reason there were a lot more police foot patrols at that time.
Another advantage of the foot patrols was that the street cops got to know who the troublemakers in the neighborhood were, and in particular, who the crazies were. Much less anonymity in those days.
@Carazaa Have you seen the old movies were a telex machine prints out a single line of a message on a strip of paper? See video at 2 minutes [media=https://youtu.be/WJrm0B-EF3o]