He said the marches, which have been regularly held on the streets of London and other cities since the Israel-Hamas conflict began, could "spill over" into hate speech and anti-Semitism.
John Zak Woodock, the government's political violence tsar, said he was concerned about the "limited" police response to the protests.
There were seven arrests as thousands of people marched in London on Saturday, including a man seen carrying a coffin with offensive language written on it and another leading a chant of "intifada revolution."
Scotland Yard said officers also arrested a 74-year-old man at a counter-protest on suspicion of provocation of violence.
Woodcock, who is due to publish a long-awaited report into political violence and disruption this week, told Sky News: "I've been concerned over many months now that the police do seem limited in what they are able to do to balance people's right to protest with the cumulative impact of having marches through central London on a weekly basis."
"This is making substantial parts of the community, sizeable parts of our Jewish community in London, apprehensive at best about going into the centre of the city."
"That is a deeply uncomfortable position."
"There has been a substantial level of criminality and disorder and antisemitic content around the margins of the marches."
Shapps told GB News: "It worries me, it worries the government a lot."
"I was pleased to see there were ten or so arrests made yesterday. We have said in the past, I think the police need to be very much all over these things."
"We see the impact on people. It makes people feel very uncomfortable when these marches sometimes spill over from the legitimate right to protest and make a point - that is not an issue - when they spill over into anti-Semitism, hate speech, racism, that is where there is a problem, and we absolutely back 100 per cent police taking the necessary action."
"I see now that they have charged hundreds and hundreds of people over the last few months, and there were a significant number of arrests yesterday, which I think was the right way to deal with these things."
In Israel, Prime Minister Benjamin Netanyahu's main political rival, Benny Gantz, has threatened to leave the war cabinet if a plan is not drawn up by June 8 that includes an international administration for post-war Gaza.
Shapps told Times Radio: "The point we constantly make to Israel is it's incredibly important we have a plan - a Gaza run not by Hamas, a terrorist organisation, but by Palestinians."
"It can't be in anyone's interest, least of all Israel's, to be in Gaza in the long run."
The Cabinet Minister also defended the UK's efforts to get aid into Gaza, although he acknowledged the new US-built pier would only have a limited impact.
He told the BBC it would be "ridiculous" to suggest the pier was the answer but added: "Does that mean we shouldn't be doing it? Not at all. Of course, we should be doing it."
"We have, the United Kingdom, without much fanfare, been delivering aid by sea, by air, by road, and the pier is just another way of getting aid in, and we're helping to organise the dispatch of that from Cyprus."
"We have the command and control ship RFA Cardigan Bay in the Mediterranean, and so we're assisting with getting that aid onto the land in Gaza.
"Of course, it's not enough. I mean, it would be ridiculous to suggest otherwise."