She told BBC Radio 4's Today programme that people should take more notice of their surroundings.
Measures such as the closure of railway stations in Munich on New Year's Eve would become more likely.
Such moves showed "prudence and proper caution" by authorities, she said.
Official guidance is that members of the public should be "alert but not alarmed" about the threat of terrorism.
But Lady Neville-Jones, former head of the Joint Intelligence Committee, said people were not as vigilant as they could be.
"I think being alert is very important," she said. "I am alarmed by the number of people I see wandering along the street entirely engaged in their mobile telephones and with their ears plugged in to music.
"They're not aware of their surroundings - you need to be aware of your surroundings."
When interviewer John Humphrys said that was "how we should be able to lead our lives", she added: "Maybe, but I do think you need to take some personal responsibility."
Two railway stations which were closed in Munich in response to a suspected terror threat reopened on Friday after German officials said there was no sign of an imminent attack.
Lady Neville-Jones said: "I do think we are going to have to get used to what we have seen yesterday in Germany - that is to say alerts which close places of public functions, and higher levels of security in big public gatherings.
"That is prudence and proper caution on the part of intelligence and the police authorities."
Authorities have a responsibility to take information seriously, even if the wider picture is not clear, she said.
"It's a very bold government or policeman who chooses not to take precautions in such circumstances," said Lady Neville-Jones.
"I think the population on the whole would prefer them to be cautious and occasionally have closed something that it turned out wasn't necessary, but how do we know, rather than take the risk of exposing people to dangers on which they have information, even if it's not complete and on which they can't necessarily totally rely."
But she said UK cities were unlikely to be locked down in the way Brussels had been over recent terrorism concerns.
She told Today: "I would hope that that would not be something that would happen in the UK.
"I do think that counter-terrorism and both the intelligence side of it and the policing side of it, and well done, are matters which are bred of long experience and of great skill and I think that in this country we do have both of those things and we have very close co-operation between both police and agencies and I think they understand how to use information.
"I don't think those skills are nearly so widespread on the continent. So I think we shouldn't conclude that what happened there is necessarily the technique that would be adopted here."

The planned hub, which is still at the consultation stage, will be at the Europa bus centre/Great Victoria Street railway station.
The two stations are the busiest in Belfast, with about eight million passengers a year passing through them.
It is expected that figure will rise to 13 million in the next 14 years.
Translink described the Belfast Hub as a "transport-led regeneration project" which will consist of a "world-class transport interchange and surrounding masterplan development".
The site would cover 20 acres from the stations to the Royal Victoria Hospital.
Half of that would be used for an integrated transport hub and the other half would be open to commercial development.
Over the past 18 months, Translink has engaged with local stakeholders to help shape the development proposals and said this latest stage "invites the public to share their views on the proposals".
Launching the public consultation, Translink group chief executive, Chris Conway, said: "The Belfast Hub will ensure we have the right infrastructure in place to attract more people to use public transport and active travel modes, supporting a key outcome of the draft programme for government.
"Along with other key projects, it will enable a transformation in public transport making a positive impact for everyone in Northern Ireland.
"Identified as one of the NI Executive's flagship projects, this impressive new facility will act as a catalyst for regeneration in one of the special action areas designated in Belfast City Council's city centre and regeneration strategy, providing new jobs, investment opportunities and supporting the growth and prosperity of NI."
The public can contribute to the consultation in the following ways:
The consultation stage is open until 2 December.
Translink said a further consultation will take place in 2017 before a planning application is submitted.
Former security minister Baroness Neville-Jones says she is "alarmed" by people using mobile phones in public instead of being more alert to the risk of a potential terrorist attack.
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Translink is consulting the public on proposals for a new transport hub in Belfast city centre.