Short answer: The method may or may not completely cure asthma but may be used to reduce dependency on an inhaler. It is not proven to work for all patients.
It has wider acceptance in Australia and UK and more opposition in the USA from what I've found so far. See the last bit of my post on that.
Proponents of the method may know that
British Guideline on the Management of Asthma - A national clinical guideline from the NHS and British Thoracic Society in 2008 (See PDF pg 35) gave it the following recommendation
The Buteyko breathing technique specifi cally focuses on control of
hyperventilation and any ensuing hypocapnia. Four clinical trials
suggest benefi ts in terms of reduced symptoms and bronchodilator
usage but no effect on lung function.
Buteyko breathing technique
may be considered to help patients to control the symptoms of asthma.
In April 2009, a course to train nurses how to teach the Buteyko Breathing Technique to people with asthma was launched by Coventry University and the Buteyko Breathing Association. This shows a more mainstream acceptance of this technique in the UK.
There have been at least 8 clinical trials as listed on this page. Notably none are in the USA and none are very large groups (none more than 69 subjects). This page also links to abstracts of each of the trials - I've summarized conclusions below.
ENGLAND 2009 (32 subjects, 5 weeks)
By teaching patients to reduce hypernoea of breathing (the rate &
depth), BBT may reduce asthma symptoms and improve exercise tolerance
CANADA 2008 (56 subjects)
Six months after completion of the interventions, a large majority of
subjects in each group displayed control of their asthma with the
additional benefit of reduction in inhaled corticosteroid use in the
NEW ZEALAND 2006 (8 subjects)
Conclusions: children’s decreased use of reliever medication mirror
adults and warrant further studies.
AUSTRALIA 2006 (23 subjects)
Conclusion: breathing techniques useful in management of asthma.
The process of routine and as required exercises may reinforce a
message of relaxation and self efficacy and provide a deferral
strategy for reliever use.
ENGLAND 2003 (23 subjects)
Conclusion: Buteyko can reduce bronchodilator use and improve
symptoms. Buteyko may be worth trying in patients who are sympathetic
to the ethos and are willing to commit the time required.
Choice of control is difficult, maintaining blindness is a problem.
Limited follow up visits to one for each group to remain comparable,
but some subjects would have liked more. Success or failure relies
heavily on patient/practitioner relationship.
Paitents need to be motivated, so the success rate of a formal trial
may be less than that seen in everyday practice, and since patients
did not chose the Buteyko couse, and because they are not paying, they
may be less committed to seeing it through.
NEW ZEALAND 2003 (17 subjects)
At six months Buteyko Group (n=17) or control (n=17) asthma education
• Reliever medication decreased by 85% in Buteyko
group, by 37% in control group.
• Inhaled corticosteroids decreased
by 50% in Buteyko group, unchanged in control.
Conclusions: Buteyko may be effective in improving quality of life and
reducing inhaled reliever medication in patients with asthma –warrants
AUSTRALIA 1998 (19 subjects)
At three months Buteyko Group achieved
• Reliever medication
decreased by 90%. Significantly more than control (P=0.002)
towards reduction of inhaled corticosteroids
So now on to the opposition, the Wikipedia article you've linked has the U.S. National Heart, Lung, and Blood Institute asthma guideline concluding this summary
The Expert Panel concludes there is insufficient evidence to suggest
that breathing techniques provide clinical benefit to patients who
have asthma. Controlled studies have been conducted with breathing
exercises (Holloway and Ram 2004), inspiratory muscle training (Ram et
al. 2003; Weiner et al. 2002), and Buteyko breathing (Cooper et al.
2003) (raising blood PCO2 through hypoventilation).
but Cooper's actual study was on Buteyko v Pranayam and notes
Symptoms remained relatively stable in the PCLE (to mimic Pranayam) and placebo groups but
were reduced in the Buteyko group
and concludes that inhaler use may be reduced
CONCLUSION: The Buteyko breathing technique can improve symptoms and
reduce bronchodilator use but does not appear to change bronchial
responsiveness or lung function in patients with asthma. No benefit
was shown for the Pink City Lung Exerciser.
From what I've seen on some forums, the US opposition is a conspiracy angle, that this will reduce sales for large pharmaceuticals.
Some further reading