The detail about the Dover Mail guard having a weapon on his person comes from Book 1, Chapter 2.
The team of horses pulling the coach are struggling to haul the vehicle up the mud-covered Shooter Hill at night in November. The three passengers traveling in the coach have had to get out and walk in order to help ease the load. During a description of the passengers, the narrator explains that none can discern the appearance of his fellow travelers because each is covered head to toe in both clothing and mud.
The travelers are pleased to be so concealed because of their suspicion. According to the text, “anybody on the road might be a robber or in league with robbers.” As a result, many travelers were wary of letting a stranger know anything about themselves.
This also explains why the guard himself feels the need to carry protection. The narrator explains that the coachman is glad to have his chest full of weapons. The coachman keeps “an eye and a hand” on the chest, which contains “a loaded blunderbuss” and “six or eight loaded horse pistols” sitting atop a pile of swords. This shows that the guard is overprepared in case of an attack.
The narrator further explains the aura of suspicion among all on the mail coach. Each person is suspicious of all the others, and the driver trusts “only the horses.” This shows that people who traveled or worked on mail coaches at the time were paranoid.
The narrator explains how the guard has an additional store of weapons or tools that could be used as weapons, just in case something horrific were to occur.
Besides the fear of vulnerability to theft, this paranoia indicates a lack of faith in one’s fellow man, an idea that continues to be addressed throughout the novel.