While pen and paper can be useful for simple DAGs, or for recording ideas when brainstorming or just thinking through a research question, more realistic DAGs generally require the use of software.
There are two broad types: software that creates diagrams like flowcharts, network diagrams, and many others; and software that is specifically designed to create DAGs. Both have their advantages and disadvantages, and many researchers use both depending on what the DAG is being created for.
General diagramming software
These are often simple to use and are good for making causal diagrams that are easy for readers to understand. These might be for research publications, or for educational purposes, such as to help people understand a type of bias. Commonly used diagramming software includes:
Visio 
PowerPoint 
LucidChart
Gliffy
Diagrams.net
Google Drawings
And many more, with new ones appearing regularly
They do not offer features that are specific to DAGs, however, such as suggesting variables to adjust for and to not adjust for.
Software with DAG-specific features
DAGitty
The most well-known software package specifically designed to create DAGs and known to health researchers through publications is DAGitty, available as a web application and also as the R package ‘dagitty’, along with the related R packages ggdag and shinyDAG .
Causal Fusion
Note: Still in beta so you will need to sign up and be granted permission
While these are not commercial packages, many health researchers use them, and it is probably worth the brief learning curve to make use of their unique functionality, even if other software is used in tandem to produce a diagram for publication.